Integral Table — Integration Formulas Reference A complete integration formula reference — the integral table students actually use. Includes basic antiderivatives, trig and inverse trig, exponential and log, plus standard techniques like u-substitution and integration by parts. The constant of integration C is implied for indefinite integrals.
Basic Integrals Power Rule
∫ x n d x = x n + 1 n + 1 + C ( n ≠ − 1 ) \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad(n \ne -1) ∫ x n d x = n + 1 x n + 1 + C ( n = − 1 ) Reciprocal
∫ 1 x d x = ln ∣ x ∣ + C \int \frac{1}{x}\,dx = \ln|x| + C ∫ x 1 d x = ln ∣ x ∣ + C Constant Multiple
∫ c f ( x ) d x = c ∫ f ( x ) d x \int c\,f(x)\,dx = c\int f(x)\,dx ∫ c f ( x ) d x = c ∫ f ( x ) d x Sum/Difference
∫ [ f ( x ) ± g ( x ) ] d x = ∫ f d x ± ∫ g d x \int [f(x) \pm g(x)]\,dx = \int f\,dx \pm \int g\,dx ∫ [ f ( x ) ± g ( x )] d x = ∫ f d x ± ∫ g d x Fundamental Theorem of Calculus
∫ a b f ′ ( x ) d x = f ( b ) − f ( a ) \int_a^b f'(x)\,dx = f(b) - f(a) ∫ a b f ′ ( x ) d x = f ( b ) − f ( a )
Exponential & Logarithmic Integrals Natural Exponential
∫ e x d x = e x + C \int e^x\,dx = e^x + C ∫ e x d x = e x + C General Exponential
∫ a x d x = a x ln a + C \int a^x\,dx = \frac{a^x}{\ln a} + C ∫ a x d x = ln a a x + C Exponential with u
∫ e u ( x ) u ′ ( x ) d x = e u ( x ) + C \int e^{u(x)} u'(x)\,dx = e^{u(x)} + C ∫ e u ( x ) u ′ ( x ) d x = e u ( x ) + C Natural Log of x
∫ ln x d x = x ln x − x + C \int \ln x\,dx = x \ln x - x + C ∫ ln x d x = x ln x − x + C Log with u
∫ u ′ ( x ) u ( x ) d x = ln ∣ u ( x ) ∣ + C \int \frac{u'(x)}{u(x)}\,dx = \ln|u(x)| + C ∫ u ( x ) u ′ ( x ) d x = ln ∣ u ( x ) ∣ + C Trigonometric Integrals Sine
∫ sin x d x = − cos x + C \int \sin x\,dx = -\cos x + C ∫ sin x d x = − cos x + C Cosine
∫ cos x d x = sin x + C \int \cos x\,dx = \sin x + C ∫ cos x d x = sin x + C Secant Squared
∫ sec 2 x d x = tan x + C \int \sec^2 x\,dx = \tan x + C ∫ sec 2 x d x = tan x + C Cosecant Squared
∫ csc 2 x d x = − cot x + C \int \csc^2 x\,dx = -\cot x + C ∫ csc 2 x d x = − cot x + C Sec Tan
∫ sec x tan x d x = sec x + C \int \sec x \tan x\,dx = \sec x + C ∫ sec x tan x d x = sec x + C Csc Cot
∫ csc x cot x d x = − csc x + C \int \csc x \cot x\,dx = -\csc x + C ∫ csc x cot x d x = − csc x + C Tangent
∫ tan x d x = − ln ∣ cos x ∣ + C \int \tan x\,dx = -\ln|\cos x| + C ∫ tan x d x = − ln ∣ cos x ∣ + C Cotangent
∫ cot x d x = ln ∣ sin x ∣ + C \int \cot x\,dx = \ln|\sin x| + C ∫ cot x d x = ln ∣ sin x ∣ + C Secant
∫ sec x d x = ln ∣ sec x + tan x ∣ + C \int \sec x\,dx = \ln|\sec x + \tan x| + C ∫ sec x d x = ln ∣ sec x + tan x ∣ + C Cosecant
∫ csc x d x = − ln ∣ csc x + cot x ∣ + C \int \csc x\,dx = -\ln|\csc x + \cot x| + C ∫ csc x d x = − ln ∣ csc x + cot x ∣ + C Inverse Trig Integrals Arcsine Form
∫ d x a 2 − x 2 = sin − 1 ( x a ) + C \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\!\left(\frac{x}{a}\right) + C ∫ a 2 − x 2 d x = sin − 1 ( a x ) + C Arctangent Form
∫ d x a 2 + x 2 = 1 a tan − 1 ( x a ) + C \int \frac{dx}{a^2 + x^2} = \frac{1}{a}\tan^{-1}\!\left(\frac{x}{a}\right) + C ∫ a 2 + x 2 d x = a 1 tan − 1 ( a x ) + C Arcsecant Form
∫ d x x x 2 − a 2 = 1 a sec − 1 ∣ x a ∣ + C \int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a}\sec^{-1}\!\left|\frac{x}{a}\right| + C ∫ x x 2 − a 2 d x = a 1 sec − 1 a x + C Integration Techniques u-Substitution
∫ f ( g ( x ) ) g ′ ( x ) d x = ∫ f ( u ) d u ( u = g ( x ) ) \int f(g(x))\,g'(x)\,dx = \int f(u)\,du \quad(u = g(x)) ∫ f ( g ( x )) g ′ ( x ) d x = ∫ f ( u ) d u ( u = g ( x )) Integration by Parts
∫ u d v = u v − ∫ v d u \int u\,dv = uv - \int v\,du ∫ u d v = uv − ∫ v d u Trig Substitution: √(a² − x²)
x = a sin θ x = a \sin\theta x = a sin θ Trig Substitution: √(a² + x²)
x = a tan θ x = a \tan\theta x = a tan θ Trig Substitution: √(x² − a²)
x = a sec θ x = a \sec\theta x = a sec θ Partial Fractions (Linear)
P ( x ) ( x − a ) ( x − b ) = A x − a + B x − b \frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} ( x − a ) ( x − b ) P ( x ) = x − a A + x − b B Special Integrals Improper Integral
∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x \int_a^\infty f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx ∫ a ∞ f ( x ) d x = b → ∞ lim ∫ a b f ( x ) d x Average Value of a Function
f avg = 1 b − a ∫ a b f ( x ) d x f_\text{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx f avg = b − a 1 ∫ a b f ( x ) d x ∃ c ∈ [ a , b ] : f ( c ) = 1 b − a ∫ a b f ( x ) d x \exists\,c \in [a,b]: f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx ∃ c ∈ [ a , b ] : f ( c ) = b − a 1 ∫ a b f ( x ) d x Gaussian Integral
∫ − ∞ ∞ e − x 2 d x = π \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi} ∫ − ∞ ∞ e − x 2 d x = π
