Table of Integral Formulas
A table of indefinite integrals of functions is presented below.
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1 - Integrals of Elementary Functions.
1.1 ∫ d x = x + c \displaystyle \int \; dx = x + c ∫ d x = x + c ∫ k d x = k x + c \displaystyle \int k \; dx = k x + c ∫ k d x = k x + c k k k ∫ x n d x = x n + 1 n + 1 + c \displaystyle \int x^n \; dx = \dfrac{x^{n+1}}{n+1} + c ∫ x n d x = n + 1 x n + 1 + c ∫ 1 x d x = ln ∣ x ∣ + c \displaystyle \int \dfrac{1}{x} \; dx = \ln |x| + c ∫ x 1 d x = ln ∣ x ∣ + c
2 - Integrals of Elementary Trigonometric Functions : sin x \sin x sin x cos x \cos x cos x tan x \tan x tan x cot x \cot x cot x sec x \sec x sec x csc x \csc x csc x
2.1 ∫ sin x d x = − cos x + c \displaystyle \int \sin x \; dx = -\cos x + c ∫ sin x d x = − cos x + c ∫ cos x d x = sin x + c \displaystyle \int \cos x \; dx = \sin x + c ∫ cos x d x = sin x + c ∫ tan x d x = ln ∣ sec x ∣ + c \displaystyle \int \tan x \; dx = \ln |\sec x| + c ∫ tan x d x = ln ∣ sec x ∣ + c ∫ cot x d x = ln ∣ sin x ∣ + c \displaystyle \int \cot x \; dx = \ln |\sin x| + c ∫ cot x d x = ln ∣ sin x ∣ + c ∫ sec x d x = ln ∣ sec x + tan x ∣ + c \displaystyle \int \sec x \; dx = \ln |\sec x + \tan x| + c ∫ sec x d x = ln ∣ sec x + tan x ∣ + c ∫ csc x d x = ln ∣ csc x − cot x ∣ + c \displaystyle \int \csc x \; dx = \ln |\csc x - \cot x| + c ∫ csc x d x = ln ∣ csc x − cot x ∣ + c
3 - Integrals Involving More Than One Trigonometric Function.
3.1 ∫ sec x tan x d x = sec x + c \displaystyle \int \sec x \tan x \; dx = \sec x + c ∫ sec x tan x d x = sec x + c ∫ csc x cot x d x = − csc x + c \displaystyle \int \csc x \cot x \; dx = - \csc x + c ∫ csc x cot x d x = − csc x + c ∫ sin m x sin n x d x = − sin [ ( m + n ) x ] 2 ( m + n ) + sin [ ( m − n ) x ] 2 ( m − n ) + c \displaystyle \int \sin mx \sin nx \; dx = -\dfrac{\sin[(m+n)x]}{2(m+n)} + \dfrac{\sin[(m-n)x]}{2(m-n)} + c ∫ sin m x sin n x d x = − 2 ( m + n ) sin [( m + n ) x ] + 2 ( m − n ) sin [( m − n ) x ] + c with m ≠ n m \neq n m = n
∫ cos m x cos n x d x = sin [ ( m + n ) x ] 2 ( m + n ) + sin [ ( m − n ) x ] 2 ( m − n ) + c \displaystyle \int \cos mx \cos nx \; dx = \dfrac{\sin[(m+n)x]}{2(m+n)} + \dfrac{\sin[(m-n)x]}{2(m-n)} + c ∫ cos m x cos n x d x = 2 ( m + n ) sin [( m + n ) x ] + 2 ( m − n ) sin [( m − n ) x ] + c with m ≠ n m \neq n m = n
∫ sin m x cos n x d x = − cos [ ( m + n ) x ] 2 ( m + n ) − cos [ ( m − n ) x ] 2 ( m − n ) + c \displaystyle \int \sin mx \cos nx \; dx = -\dfrac{\cos[(m+n)x]}{2(m+n)} - \dfrac{\cos[(m-n)x]}{2(m-n)} + c ∫ sin m x cos n x d x = − 2 ( m + n ) cos [( m + n ) x ] − 2 ( m − n ) cos [( m − n ) x ] + c with m ≠ n m \neq n m = n
4 - Integrals Involving Exponential and Logarithmic Functions.
4.1 ∫ e x d x = e x + c \displaystyle \int e^x \; dx = e^x + c ∫ e x d x = e x + c ∫ a x d x = a x ln a + c \displaystyle \int a^x \; dx = \dfrac{a^x}{\ln a} + c ∫ a x d x = ln a a x + c ∫ ln x d x = x ln x − x + c \displaystyle \int \ln x \; dx = x \ln x - x + c ∫ ln x d x = x ln x − x + c
5 - Integrals of Inverse Trigonometric functions: arcsin x \arcsin x arcsin x arccos x \arccos x arccos x arctan x \arctan x arctan x arccot x \text{arccot} \; x arccot x arcsec x \text{arcsec}\; x arcsec x arccsc x \text{arccsc} \; x arccsc x
5.1 ∫ arcsin x d x = x arcsin x + 1 − x 2 + c \displaystyle \int \arcsin x \; dx = x \arcsin x + \sqrt{1 - x^2} + c ∫ arcsin x d x = x arcsin x + 1 − x 2 + c ∫ arccos x d x = x arccos x − 1 − x 2 + c \displaystyle \int \arccos x \; dx = x \arccos x - \sqrt{1 - x^2} + c ∫ arccos x d x = x arccos x − 1 − x 2 + c ∫ arctan x d x = x arctan x − ln ∣ 1 + x 2 ∣ + c \displaystyle \int \arctan x \; dx = x \arctan x - \ln \left| \sqrt{1 + x^2} \right| + c ∫ arctan x d x = x arctan x − ln 1 + x 2 + c ∫ arccot x d x = x arccot x + ln 1 + x 2 + c \displaystyle \int \text{arccot} \; x \; dx = x \; \text{arccot} \; x + \ln \sqrt{1 + x^2} + c ∫ arccot x d x = x arccot x + ln 1 + x 2 + c ∫ arcsec x d x = x arcsec x − ln ∣ x + x 2 − 1 ∣ + c \displaystyle \int \text{arcsec} \; x \; dx = x \; \text{arcsec} \; x - \ln \left| x + \sqrt{x^2 - 1} \right| + c ∫ arcsec x d x = x arcsec x − ln x + x 2 − 1 + c ∫ arccsc x d x = x arccsc x + ln ∣ x + x 2 − 1 ∣ + c \displaystyle \int \text{ arccsc} \; x \; dx = x \; \text{ arccsc} \; x + \ln \left| x + \sqrt{x^2 - 1} \right| + c ∫ arccsc x d x = x arccsc x + ln x + x 2 − 1 + c
6 - Integrals Involving Exponential and Sine and Cosine Functions.
6.1 ∫ e a x sin b x d x = e a x a 2 + b 2 ( a sin b x − b cos b x ) + c \displaystyle \int e^{ax} \sin bx \; dx = \dfrac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx) + c ∫ e a x sin b x d x = a 2 + b 2 e a x ( a sin b x − b cos b x ) + c ∫ e a x cos b x d x = e a x a 2 + b 2 ( b sin b x + a cos b x ) + c \displaystyle \int e^{ax} \cos bx \; dx = \dfrac{e^{ax}}{a^2 + b^2} (b \sin bx + a \cos bx) + c ∫ e a x cos b x d x = a 2 + b 2 e a x ( b sin b x + a cos b x ) + c
7 - Integrals Involving Hyperbolic Functions: sinh x \sinh x sinh x cosh x \cosh x cosh x tanh x \tanh x tanh x coth x \coth x coth x sech x \text{sech} \; x sech x csch x \text{csch} \; x csch x
7.1 ∫ sinh x d x = cosh x + c \displaystyle \int \sinh x \; dx = \cosh x + c ∫ sinh x d x = cosh x + c ∫ cosh x d x = sinh x + c \displaystyle \int \cosh x \; dx = \sinh x + c ∫ cosh x d x = sinh x + c ∫ sech x tanh x d x = − sech x + c \displaystyle \int \text{sech} \; x \tanh x \; dx = - \; \text{sech} \; x + c ∫ sech x tanh x d x = − sech x + c ∫ csch x coth x d x = − csch x + c \displaystyle \int \text{csch} \; x \coth x \; dx = - \; \text{csch} \; x + c ∫ csch x coth x d x = − csch x + c ∫ sech 2 x d x = tanh x + c \displaystyle \int \text{sech}^2 x \; dx = \tanh x + c ∫ sech 2 x d x = tanh x + c ∫ csch 2 x d x = − coth x + c \displaystyle \int \text{csch}^2 x \; dx = - \;\coth x + c ∫ csch 2 x d x = − coth x + c integrales y sus aplicaciones en cálculo.