Integral formulas The integral formulas you use most, from basic antiderivatives to substitution, integration by parts, trig and rational functions, and properties of definite integrals. Each section links to practice problems.
Basic antiderivatives Linearity and linear inner functions Substitution and integration by parts Trigonometric integrals and identities Rational and irrational functions Definite integrals Area and volume Improper and double integrals
Basic antiderivatives
Below, C C C log \log log ln \ln ln
Function Indefinite integral Note x n x^n x n x n + 1 n + 1 + C \dfrac{x^{n+1}}{n+1} + C n + 1 x n + 1 + C n = 0 , 1 , 2 , … n = 0, 1, 2, \ldots n = 0 , 1 , 2 , … x α x^\alpha x α x α + 1 α + 1 + C \dfrac{x^{\alpha+1}}{\alpha+1} + C α + 1 x α + 1 + C α ≠ − 1 \alpha \ne -1 α = − 1 1 x \dfrac{1}{x} x 1 log ∣ x ∣ + C \log|x| + C log ∣ x ∣ + C Keep the absolute value e x e^x e x e x + C e^x + C e x + C a x a^x a x a x log a + C \dfrac{a^x}{\log a} + C log a a x + C a > 0 , a ≠ 1 a > 0,\ a \ne 1 a > 0 , a = 1 sin x \sin x sin x − cos x + C -\cos x + C − cos x + C cos x \cos x cos x sin x + C \sin x + C sin x + C 1 cos 2 x \dfrac{1}{\cos^2 x} cos 2 x 1 tan x + C \tan x + C tan x + C 1 sin 2 x \dfrac{1}{\sin^2 x} sin 2 x 1 − 1 tan x + C -\dfrac{1}{\tan x} + C − tan x 1 + C tan x \tan x tan x − log ∣ cos x ∣ + C -\log|\cos x| + C − log ∣ cos x ∣ + C By substitution log x \log x log x x log x − x + C x\log x - x + C x log x − x + C By parts
→ Practice: indefinite integrals of polynomials
→ Practice: basic indefinite integrals
Linearity and linear inner functions
Function Indefinite integral ( a x + b ) n (ax+b)^n ( a x + b ) n ( a x + b ) n + 1 a ( n + 1 ) + C \dfrac{(ax+b)^{n+1}}{a(n+1)} + C a ( n + 1 ) ( a x + b ) n + 1 + C n ≠ − 1 n \ne -1 n = − 1 1 a x + b \dfrac{1}{ax+b} a x + b 1 1 a log ∣ a x + b ∣ + C \dfrac{1}{a}\log|ax+b| + C a 1 log ∣ a x + b ∣ + C e a x + b e^{ax+b} e a x + b 1 a e a x + b + C \dfrac{1}{a}e^{ax+b} + C a 1 e a x + b + C sin ( a x + b ) \sin(ax+b) sin ( a x + b ) − 1 a cos ( a x + b ) + C -\dfrac{1}{a}\cos(ax+b) + C − a 1 cos ( a x + b ) + C cos ( a x + b ) \cos(ax+b) cos ( a x + b ) 1 a sin ( a x + b ) + C \dfrac{1}{a}\sin(ax+b) + C a 1 sin ( a x + b ) + C
Substitution and integration by parts
Function Indefinite integral x e x xe^x x e x ( x − 1 ) e x + C (x-1)e^x + C ( x − 1 ) e x + C x 2 e x x^2e^x x 2 e x ( x 2 − 2 x + 2 ) e x + C (x^2-2x+2)e^x + C ( x 2 − 2 x + 2 ) e x + C By parts twice x sin x x\sin x x sin x − x cos x + sin x + C -x\cos x + \sin x + C − x cos x + sin x + C x cos x x\cos x x cos x x sin x + cos x + C x\sin x + \cos x + C x sin x + cos x + C x log x x\log x x log x x 2 2 log x − x 2 4 + C \dfrac{x^2}{2}\log x - \dfrac{x^2}{4} + C 2 x 2 log x − 4 x 2 + C e a x sin b x e^{ax}\sin bx e a x sin b x e a x ( a sin b x − b cos b x ) a 2 + b 2 + C \dfrac{e^{ax}(a\sin bx - b\cos bx)}{a^2+b^2} + C a 2 + b 2 e a x ( a sin b x − b cos b x ) + C Cyclic e a x cos b x e^{ax}\cos bx e a x cos b x e a x ( a cos b x + b sin b x ) a 2 + b 2 + C \dfrac{e^{ax}(a\cos bx + b\sin bx)}{a^2+b^2} + C a 2 + b 2 e a x ( a cos b x + b sin b x ) + C Cyclic
→ Practice: integration by substitution
→ Practice: integration by parts
Trigonometric integrals and identities
Function Indefinite integral sin 2 x \sin^2 x sin 2 x x 2 − sin 2 x 4 + C \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C 2 x − 4 sin 2 x + C cos 2 x \cos^2 x cos 2 x x 2 + sin 2 x 4 + C \dfrac{x}{2} + \dfrac{\sin 2x}{4} + C 2 x + 4 sin 2 x + C tan 2 x \tan^2 x tan 2 x tan x − x + C \tan x - x + C tan x − x + C sin 3 x \sin^3 x sin 3 x − cos x + 1 3 cos 3 x + C -\cos x + \dfrac{1}{3}\cos^3 x + C − cos x + 3 1 cos 3 x + C 1 cos x \dfrac{1}{\cos x} cos x 1 1 2 log 1 + sin x 1 − sin x + C \dfrac{1}{2}\log\dfrac{1+\sin x}{1-\sin x} + C 2 1 log 1 − sin x 1 + sin x + C
→ Practice: trigonometric integrals
Rational and irrational functions
Function Indefinite integral Note 1 ( x + a ) ( x + b ) \dfrac{1}{(x+a)(x+b)} ( x + a ) ( x + b ) 1 1 b − a log ∣ x + a x + b ∣ + C \dfrac{1}{b-a}\log\left|\dfrac{x+a}{x+b}\right| + C b − a 1 log x + b x + a + C a ≠ b a \ne b a = b 1 x 2 − a 2 \dfrac{1}{x^2-a^2} x 2 − a 2 1 1 2 a log ∣ x − a x + a ∣ + C \dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right| + C 2 a 1 log x + a x − a + C a > 0 a > 0 a > 0 1 x 2 + a 2 \dfrac{1}{x^2+a^2} x 2 + a 2 1 1 a arctan x a + C \dfrac{1}{a}\arctan\dfrac{x}{a} + C a 1 arctan a x + C University 1 a 2 − x 2 \dfrac{1}{\sqrt{a^2-x^2}} a 2 − x 2 1 arcsin x a + C \arcsin\dfrac{x}{a} + C arcsin a x + C University 1 x 2 + A \dfrac{1}{\sqrt{x^2+A}} x 2 + A 1 log ∣ x + x 2 + A ∣ + C \log\left|x+\sqrt{x^2+A}\right| + C log x + x 2 + A + C University a 2 − x 2 \sqrt{a^2-x^2} a 2 − x 2 1 2 ( x a 2 − x 2 + a 2 arcsin x a ) + C \dfrac{1}{2}\left(x\sqrt{a^2-x^2}+a^2\arcsin\dfrac{x}{a}\right) + C 2 1 ( x a 2 − x 2 + a 2 arcsin a x ) + C University a x + b \sqrt{ax+b} a x + b 2 3 a ( a x + b ) a x + b + C \dfrac{2}{3a}(ax+b)\sqrt{ax+b} + C 3 a 2 ( a x + b ) a x + b + C
→ Practice: rational functions (partial fractions)
→ Practice: integrals with inverse trig functions
Improper and double integrals
Integral Value Condition ∫ 1 ∞ d x x p \displaystyle\int_1^\infty \frac{dx}{x^p} ∫ 1 ∞ x p d x 1 p − 1 \dfrac{1}{p-1} p − 1 1 Converges for p > 1 p > 1 p > 1 ∫ 0 1 d x x p \displaystyle\int_0^1 \frac{dx}{x^p} ∫ 0 1 x p d x 1 1 − p \dfrac{1}{1-p} 1 − p 1 Converges for p < 1 p < 1 p < 1 ∫ 0 ∞ x n e − x d x \displaystyle\int_0^\infty x^n e^{-x}\,dx ∫ 0 ∞ x n e − x d x n ! n! n ! Gamma function ∫ − ∞ ∞ e − x 2 d x \displaystyle\int_{-\infty}^{\infty} e^{-x^2}\,dx ∫ − ∞ ∞ e − x 2 d x π \sqrt{\pi} π Gaussian integral ∫ 0 ∞ e − a x sin b x d x \displaystyle\int_0^\infty e^{-ax}\sin bx\,dx ∫ 0 ∞ e − a x sin b x d x b a 2 + b 2 \dfrac{b}{a^2+b^2} a 2 + b 2 b a > 0 a > 0 a > 0
→ Practice: improper integrals
→ Practice: double integrals