Integral Table

Browse common integration formulas in a searchable table of integrals. Find standard antiderivatives, check important conditions, and see how to verify results by differentiation.

Table of Common Integrals

An integral table pairs common functions with their antiderivatives. Search for a function or browse by category, then check the conditions attached to the formula.

For indefinite integrals, every answer includes an arbitrary constant of integration, written as + C.

Real variables and principal inverse functions are used throughout. exp means the natural exponential, ln means natural logarithm, abs means absolute value, and ^ denotes a power. Each formula holds on connected intervals within its conditions; constants can differ between disconnected intervals. The integration-by-parts row is a method identity, not a solved integral.

70 of 70 entries

Integrand / Antiderivative / Conditions
Name	Integrand f(x)	Antiderivative F(x) + C	Conditions
ConstantBasic Integrals	c	c·x + C	c is a real constant; x ∈ ℝ.
Details & verification: Constant The derivative of cx is c. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Power ruleBasic Integrals	x^n	x^(n+1)/(n+1) + C	n ≠ −1; x > 0 for arbitrary real n. Integer powers extend on intervals where defined.
Details & verification: Power rule The power-rule factor n + 1 cancels the denominator. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
ReciprocalBasic Integrals	1/x	ln(abs(x)) + C	x ≠ 0; constants may differ on the two intervals.
Details & verification: Reciprocal On x > 0 differentiate ln x; on x < 0 differentiate ln(−x). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Square rootBasic Integrals	√(x)	2·x^(3/2)/3 + C	x > 0.
Details & verification: Square root (2/3)(3/2)x¹ᐟ² = √x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Reciprocal square rootBasic Integrals	1/√(x)	2·√(x) + C	x > 0.
Details & verification: Reciprocal square root 2/(2√x) = 1/√x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Reciprocal squarePowers and Rational Functions	1/x^2	-1/x + C	x ≠ 0.
Details & verification: Reciprocal square The two negative signs in d(−x⁻¹)/dx cancel. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Linear denominatorPowers and Rational Functions	1/(a·x+b)	ln(abs(a·x+b))/a + C	a ≠ 0; ax + b ≠ 0.
Details & verification: Linear denominator The inner derivative a cancels the prefactor 1/a. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Linear powerCommon Substitution Patterns	(a·x+b)^n	(a·x+b)^(n+1)/(a·(n+1)) + C	a ≠ 0; n ≠ −1; ax + b > 0 for arbitrary real n.
Details & verification: Linear power Chain rule produces a(n + 1), canceling both denominator factors. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Numerator x over linearPowers and Rational Functions	x/(a·x+b)	x/a-b·ln(abs(a·x+b))/a^2 + C	a ≠ 0; ax + b ≠ 0.
Details & verification: Numerator x over linear The derivative is 1/a − b/[a(ax+b)] = x/(ax+b). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Product of linear denominatorsPowers and Rational Functions	1/(x·(a·x+b))	ln(abs(x/(a·x+b)))/b + C	b ≠ 0; x ≠ 0; ax + b ≠ 0.
Details & verification: Product of linear denominators Logarithmic differentiation gives (1/b)(1/x − a/(ax+b)). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Quadratic logarithmPowers and Rational Functions	x/(x^2+a^2)	ln(x^2+a^2)/2 + C	a > 0; x ∈ ℝ.
Details & verification: Quadratic logarithm The derivative of x² + a² is 2x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Difference of squaresPowers and Rational Functions	1/(x^2-a^2)	ln(abs((x-a)/(x+a)))/(2·a) + C	a > 0; x ≠ ±a.
Details & verification: Difference of squares The derivative is [1/(x−a) − 1/(x+a)]/(2a). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Reversed difference of squaresPowers and Rational Functions	1/(a^2-x^2)	ln(abs((a+x)/(a-x)))/(2·a) + C	a > 0; x ≠ ±a.
Details & verification: Reversed difference of squares The two logarithmic derivatives combine to 1/(a² − x²). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Squared quadraticPowers and Rational Functions	1/(x^2+a^2)^2	x/(2·a^2·(x^2+a^2))+atan(x/a)/(2·a^3) + C	a > 0; x ∈ ℝ.
Details & verification: Squared quadratic Differentiate the rational term with the quotient rule; add the arctangent derivative to cancel x². Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Quadratic denominator squaredPowers and Rational Functions	x/(x^2+a^2)^2	-1/(2·(x^2+a^2)) + C	a > 0; x ∈ ℝ.
Details & verification: Quadratic denominator squared Chain rule gives (−1/2)(−1)(2x)/(x²+a²)². Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Natural exponentialExponential Functions	exp(x)	exp(x) + C	x ∈ ℝ.
Details & verification: Natural exponential The derivative of exp x is exp x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Base-a exponentialExponential Functions	a^x	a^x/ln(a) + C	a > 0, a ≠ 1.
Details & verification: Base-a exponential Differentiation supplies ln a, canceling the denominator. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Linear exponentialExponential Functions	exp(a·x+b)	exp(a·x+b)/a + C	a ≠ 0; b real.
Details & verification: Linear exponential Chain rule supplies the factor a. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Exponential times sineExponential Functions	exp(a·x)·sin(b·x)	exp(a·x)·(a·sin(b·x)-b·cos(b·x))/(a^2+b^2) + C	a and b real; a² + b² > 0; radians.
Details & verification: Exponential times sine Product rule cancels the cosine terms; the sine coefficient becomes a² + b². Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Exponential times cosineExponential Functions	exp(a·x)·cos(b·x)	exp(a·x)·(a·cos(b·x)+b·sin(b·x))/(a^2+b^2) + C	a and b real; a² + b² > 0; radians.
Details & verification: Exponential times cosine Product rule cancels the sine terms; the cosine coefficient becomes a² + b². Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
x times exponentialExponential Functions	x·exp(a·x)	exp(a·x)·(a·x-1)/a^2 + C	a ≠ 0.
Details & verification: x times exponential Product rule gives exp(ax)[a(ax−1)+a]/a² = x exp(ax). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
LogarithmLogarithmic Functions	ln(abs(x))	x·ln(abs(x))-x + C	x ≠ 0.
Details & verification: Logarithm Product rule yields ln\|x\| + 1 − 1. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
x times logarithmLogarithmic Functions	x·ln(abs(x))	x^2·ln(abs(x))/2-x^2/4 + C	x ≠ 0.
Details & verification: x times logarithm Product rule gives x ln\|x\| + x/2 − x/2. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Squared logarithmLogarithmic Functions	ln(abs(x))^2	x·(ln(abs(x))^2-2·ln(abs(x))+2) + C	x ≠ 0.
Details & verification: Squared logarithm Differentiate the product; the extra 2ln\|x\| − 2 terms cancel. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Reciprocal logarithmic derivativeLogarithmic Functions	1/(x·ln(abs(x)))	ln(abs(ln(abs(x)))) + C	x ≠ 0 and \|x\| ≠ 1.
Details & verification: Reciprocal logarithmic derivative Two logarithmic chain factors give 1/[x ln\|x\|]. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
SineTrigonometric Integrals	sin(x)	-cos(x) + C	All real x. Radians.
Details & verification: Sine d(−cos x)/dx = sin x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
CosineTrigonometric Integrals	cos(x)	sin(x) + C	All real x. Radians.
Details & verification: Cosine d sin x/dx = cos x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
TangentTrigonometric Integrals	tan(x)	-ln(abs(cos(x))) + C	cos x ≠ 0. Radians.
Details & verification: Tangent The derivative is −(−sin x)/cos x = tan x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
CotangentTrigonometric Integrals	cot(x)	ln(abs(sin(x))) + C	sin x ≠ 0. Radians.
Details & verification: Cotangent The logarithmic derivative of sin x is cot x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
SecantTrigonometric Integrals	sec(x)	ln(abs(sec(x)+tan(x))) + C	cos x ≠ 0. Radians.
Details & verification: Secant The logarithmic derivative is (sec x tan x + sec²x)/(sec x + tan x) = sec x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
CosecantTrigonometric Integrals	csc(x)	ln(abs(csc(x)-cot(x))) + C	sin x ≠ 0. Radians.
Details & verification: Cosecant The logarithmic derivative is (−csc x cot x + csc²x)/(csc x − cot x) = csc x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Secant squaredTrigonometric Integrals	sec(x)^2	tan(x) + C	cos x ≠ 0. Radians.
Details & verification: Secant squared d tan x/dx = sec²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Cosecant squaredTrigonometric Integrals	csc(x)^2	-cot(x) + C	sin x ≠ 0. Radians.
Details & verification: Cosecant squared d(−cot x)/dx = csc²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Secant tangentTrigonometric Integrals	sec(x)·tan(x)	sec(x) + C	cos x ≠ 0. Radians.
Details & verification: Secant tangent The reciprocal derivative of cos x is sec x tan x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Cosecant cotangentTrigonometric Integrals	csc(x)·cot(x)	-csc(x) + C	sin x ≠ 0. Radians.
Details & verification: Cosecant cotangent The negative derivative of csc x is csc x cot x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Sine squaredTrigonometric Integrals	sin(x)^2	x/2-sin(2·x)/4 + C	All real x. Radians.
Details & verification: Sine squared Derivative (1 − cos 2x)/2 equals sin²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Cosine squaredTrigonometric Integrals	cos(x)^2	x/2+sin(2·x)/4 + C	All real x. Radians.
Details & verification: Cosine squared Derivative (1 + cos 2x)/2 equals cos²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Tangent squaredTrigonometric Integrals	tan(x)^2	tan(x)-x + C	cos x ≠ 0. Radians.
Details & verification: Tangent squared sec²x − 1 = tan²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Cotangent squaredTrigonometric Integrals	cot(x)^2	-cot(x)-x + C	sin x ≠ 0. Radians.
Details & verification: Cotangent squared csc²x − 1 = cot²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Sine cosine productTrigonometric Integrals	sin(x)·cos(x)	sin(x)^2/2 + C	All real x. Radians.
Details & verification: Sine cosine product The chain rule gives sin x cos x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Sine cubedTrigonometric Integrals	sin(x)^3	-cos(x)+cos(x)^3/3 + C	All real x. Radians.
Details & verification: Sine cubed Differentiate to get sin x(1 − cos²x) = sin³x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Cosine cubedTrigonometric Integrals	cos(x)^3	sin(x)-sin(x)^3/3 + C	All real x. Radians.
Details & verification: Cosine cubed Differentiate to get cos x(1 − sin²x) = cos³x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Secant cubedTrigonometric Integrals	sec(x)^3	(sec(x)·tan(x)+ln(abs(sec(x)+tan(x))))/2 + C	cos x ≠ 0. Radians.
Details & verification: Secant cubed The derivative is [sec x tan²x + sec³x + sec x]/2 = sec³x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Cosecant cubedTrigonometric Integrals	csc(x)^3	(-csc(x)·cot(x)+ln(abs(csc(x)-cot(x))))/2 + C	sin x ≠ 0. Radians.
Details & verification: Cosecant cubed The derivative is [csc x cot²x + csc³x + csc x]/2 = csc³x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
x times sineTrigonometric Integrals	x·sin(x)	sin(x)-x·cos(x) + C	All real x. Radians.
Details & verification: x times sine Product rule cancels the cosine terms. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
x times cosineTrigonometric Integrals	x·cos(x)	x·sin(x)+cos(x) + C	All real x. Radians.
Details & verification: x times cosine Product rule cancels the sine terms. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Arcsine patternInverse Trigonometric Forms	1/√(a^2-x^2)	asin(x/a) + C	a > 0; \|x\| < a; principal arcsine.
Details & verification: Arcsine pattern Chain rule yields (1/a)/√(1 − x²/a²) = 1/√(a² − x²). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Arctangent patternInverse Trigonometric Forms	1/(a^2+x^2)	atan(x/a)/a + C	a > 0; x ∈ ℝ; principal arctangent.
Details & verification: Arctangent pattern The two factors 1/a give 1/[a²(1 + x²/a²)]. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Arcsecant patternInverse Trigonometric Forms	1/(x·√(x^2-a^2))	asec(abs(x)/a)/a + C	a > 0; \|x\| > a; arcsec z = arccos(1/z).
Details & verification: Arcsecant pattern On either interval the derivative of \|x\| supplies sign(x), giving 1/[x√(x²−a²)]. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Arctangent functionInverse Trigonometric Forms	atan(x)	x·atan(x)-ln(1+x^2)/2 + C	x ∈ ℝ; principal arctangent.
Details & verification: Arctangent function Product rule extra x/(1+x²) cancels the logarithmic derivative. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Arcsine functionInverse Trigonometric Forms	asin(x)	x·asin(x)+√(1-x^2) + C	\|x\| < 1; principal arcsine.
Details & verification: Arcsine function Product rule extra x/√(1−x²) cancels the square-root derivative. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic sineHyperbolic Integrals	sinh(x)	cosh(x) + C	x ∈ ℝ.
Details & verification: Hyperbolic sine The derivative of cosh x is sinh x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic cosineHyperbolic Integrals	cosh(x)	sinh(x) + C	x ∈ ℝ.
Details & verification: Hyperbolic cosine The derivative of sinh x is cosh x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic tangentHyperbolic Integrals	tanh(x)	ln(cosh(x)) + C	x ∈ ℝ; cosh x > 0.
Details & verification: Hyperbolic tangent The logarithmic derivative is sinh x/cosh x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic cotangentHyperbolic Integrals	coth(x)	ln(abs(sinh(x))) + C	x ≠ 0.
Details & verification: Hyperbolic cotangent The logarithmic derivative is cosh x/sinh x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic secantHyperbolic Integrals	sech(x)	atan(sinh(x)) + C	x ∈ ℝ.
Details & verification: Hyperbolic secant cosh x/(1 + sinh²x) = 1/cosh x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic cosecantHyperbolic Integrals	csch(x)	ln(abs(tanh(x/2))) + C	x ≠ 0.
Details & verification: Hyperbolic cosecant The logarithmic derivative is sech²(x/2)/(2 tanh(x/2)) = 1/sinh x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic secant squaredHyperbolic Integrals	sech(x)^2	tanh(x) + C	x ∈ ℝ.
Details & verification: Hyperbolic secant squared d tanh x/dx = sech²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Hyperbolic cosecant squaredHyperbolic Integrals	csch(x)^2	-coth(x) + C	x ≠ 0.
Details & verification: Hyperbolic cosecant squared d(−coth x)/dx = csch²x. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Positive quadratic radical reciprocalRadical Integrals	1/√(x^2+a^2)	ln(x+√(x^2+a^2)) + C	a > 0; x ∈ ℝ; logarithm argument is positive.
Details & verification: Positive quadratic radical reciprocal The derivative is (1 + x/√(x²+a²))/(x + √(x²+a²)) = 1/√(x²+a²). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Difference radical reciprocalRadical Integrals	1/√(x^2-a^2)	ln(abs(x+√(x^2-a^2))) + C	a > 0; \|x\| > a; either connected interval.
Details & verification: Difference radical reciprocal The same logarithmic cancellation works on both sides with absolute values. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Semicircle radicalRadical Integrals	√(a^2-x^2)	(x·√(a^2-x^2)+a^2·asin(x/a))/2 + C	a > 0; \|x\| < a.
Details & verification: Semicircle radical Product and chain rules combine to [(a²−2x²)+a²]/[2√(a²−x²)]. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Positive quadratic radicalRadical Integrals	√(x^2+a^2)	(x·√(x^2+a^2)+a^2·ln(x+√(x^2+a^2)))/2 + C	a > 0; x ∈ ℝ.
Details & verification: Positive quadratic radical Differentiate the product and logarithm; combine over 2√(x²+a²). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Difference radicalRadical Integrals	√(x^2-a^2)	(x·√(x^2-a^2)-a^2·ln(abs(x+√(x^2-a^2))))/2 + C	a > 0; \|x\| > a.
Details & verification: Difference radical The derivative numerator is (2x²−a²) − a² = 2(x²−a²). Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Radical derivative patternRadical Integrals	x/√(x^2+a^2)	√(x^2+a^2) + C	a > 0; x ∈ ℝ.
Details & verification: Radical derivative pattern Chain rule gives 2x/[2√(x²+a²)]. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Quadratic three-halves reciprocalRadical Integrals	1/(x^2+a^2)^(3/2)	x/(a^2·√(x^2+a^2)) + C	a > 0; x ∈ ℝ.
Details & verification: Quadratic three-halves reciprocal The quotient derivative leaves a² in the numerator, canceling the prefactor. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Logarithmic substitutionCommon Substitution Patterns	u′(x)/u(x)	ln\|u(x)\| + C	u differentiable and nonzero on the interval.
Details & verification: Logarithmic substitution The logarithmic chain rule gives u′/u. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Exponential substitutionCommon Substitution Patterns	u′(x) exp(u(x))	exp(u(x)) + C	u differentiable.
Details & verification: Exponential substitution The exponential chain rule gives exp(u)u′. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Sine substitutionCommon Substitution Patterns	u′(x) cos(u(x))	sin(u(x)) + C	u differentiable; radians.
Details & verification: Sine substitution The sine chain rule gives cos(u)u′. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.
Integration by partsIntegration Properties	u(x) v′(x)	u(x)v(x) − ∫ v(x)u′(x) dx + C	u and v continuously differentiable on the interval. The remaining integral must exist.
Details & verification: Integration by parts Differentiate: (uv)′ − vu′ = uv′. This is a method identity, not a closed antiderivative. Reference formula or deriving rule. The derivative check applies on each connected interval satisfying the conditions; constants may differ between disconnected intervals.

Basic Integration Formulas

The power rule for integration states:

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1.

The excluded case has a different formula:

∫ 1/x dx = ln|x| + C, for x ≠ 0.

These formulas form the basis of many elementary integration problems.

Trigonometric Integral Table

Common trigonometric antiderivatives include:

∫ sin(x) dx = −cos(x) + C

∫ cos(x) dx = sin(x) + C

∫ sec²(x) dx = tan(x) + C

Check the relevant domain restrictions for integrals involving tangent, secant, and other functions with singularities.

Exponential and Logarithmic Integrals

The exponential function eˣ is its own antiderivative:

∫ eˣ dx = eˣ + C.

For a positive constant a other than 1:

∫ aˣ dx = aˣ/ln(a) + C.

Logarithmic integrals may require absolute values and careful attention to their domains.

How to Use an Integral Table

Start by simplifying the integrand and identifying a familiar pattern. Search for the closest matching entry, check the parameter restrictions, and apply substitution or integration by parts if necessary.

After finding an antiderivative, differentiate it to confirm that it gives the original integrand on the relevant interval.

Indefinite vs. Definite Integrals

An indefinite integral gives a family of antiderivatives and includes the constant C.

A definite integral evaluates a signed accumulation between specified bounds. When an antiderivative F exists and the Fundamental Theorem of Calculus applies, the definite integral from a to b equals F(b) − F(a).

Step-by-Step Integration Examples

Example 1 — Power rule

∫ (3x² + 2x) dx = x³ + x² + C.

Differentiate the answer to get 3x² + 2x.

Example 2 — Substitution

∫ 2x cos(x²) dx.

Let u = x². Then du = 2x dx.

The integral becomes:

∫ cos(u) du = sin(u) + C = sin(x²) + C.

Example 3 — Definite integral

∫ from 0 to 2 of 3x² dx = [x³] from 0 to 2 = 8.

Example 4 — Integration by parts

∫ x eˣ dx.

Using integration by parts:

∫ x eˣ dx = x eˣ − ∫ eˣ dx = (x − 1)eˣ + C.

Frequently Asked Questions

What is an integral table?

An integral table is a reference that lists common functions and their antiderivatives. It helps you recognize standard integration patterns and apply known formulas.

How do I use a table of integrals?

Simplify the integrand, find a matching formula, check its conditions, and apply any necessary substitution or integration rule. Differentiate your answer to verify it.

What does + C mean in an indefinite integral?

The constant C represents an arbitrary constant of integration. Functions that differ by a constant have the same derivative on a connected interval.

What is the integral of 1/x?

The indefinite integral is ln|x| + C for x ≠ 0. The absolute value allows the expression to work on both positive and negative intervals, with potentially different constants on the two intervals.

What is the integral of sin(x)?

The indefinite integral of sin(x) is −cos(x) + C because the derivative of −cos(x) is sin(x).

What is the difference between definite and indefinite integrals?

An indefinite integral gives a family of antiderivatives. A definite integral evaluates accumulation over a specified interval and produces a number when the integral exists.

Can every integral be solved using a table?

No. A table contains standard forms, but some expressions require substitution, partial fractions, integration by parts, special functions, numerical methods, or have no elementary antiderivative.

How can I check an integration formula?

Differentiate the proposed antiderivative. If the result equals the original integrand throughout the stated interval and the formula’s conditions are satisfied, the antiderivative is valid there.

Sources and verification

OpenStax — Table of Integrals and Lamar University — Common Derivatives and Integrals are reference sources for standard identities and deriving rules. Explanations are written for the individual formulas. Each antiderivative is checked by independent differentiation during development, with interval and branch restrictions stated beside the formula.

Derivative Table helps check antiderivatives. The Laplace Transform Table covers a different kind of integral transform. Browse all tables or read our methodology.