Table of Contents

Formulas and Rules for Integrals in Calculus

\( \)\( \)\( \) In what follows, \( c \) is the constant of integration.

Formulas

\( f(x) \)

\( \displaystyle \int f(x) dx \)

\( x^n \) \( \dfrac{x^{n+1}}{n+1} + c\)
\( \dfrac{1}{x} \) \( \ln |x| + c \)
\( e^x \) \( e^x + c \)
\( \ln x \) \( x \ln x - x + c \)
\( \sin x \) \( -\cos x + c \)
\( \cos x \) \( \sin x + c \)
\( \tan x \) \( -\ln |\cos x| + c \)
\( \cot x \) \( \ln |\sin x| + c \)
\( \sec x \) \( \ln( \sec x + \tan x ) + c \)
\( \csc x \) \( \ln(\csc x - \cot x) + c \)
\( \sinh x \) \( \cosh x + c \)
\( \cosh x \) \( \sinh x + c \)
\( \tanh x \) \( \ln( \cosh x) + c \)
\( \coth x \) \( \ln( \sinh x) + c \)
\( \text{sech} \; x \) \( 2 \tan^{-1}(e^x) + c \)
\( \text{csch} \; x \) \( -\ln (\coth x + \text{csch}\; x) + c \)
\( \dfrac{1}{\sqrt{a^2 - x^2}} \) \( \sin^{-1} \left(\dfrac{x}{a}\right) + c \; , |x| \lt a \)
\( \dfrac{1}{\sqrt{a^2 - x^2}} \) \( - \cos^{-1} \left(\dfrac{x}{a}\right) + c \; , |x| \lt a \)
\( \dfrac{1}{\sqrt{x^2 + a^2}} \) \( \ln(x+\sqrt{x^2 + a^2}) + c \)
\( \dfrac{1}{\sqrt{x^2 - a^2}} \) \( \ln(x+\sqrt{x^2 - a^2}) + c \)
\( \dfrac{1}{x^2 + a^2} \) \( \dfrac{1}{a} \tan^{-1} \left(\dfrac{x}{a} \right) + c \)
\( \dfrac{1}{x^2 - a^2} \) \( \dfrac{1}{2 a} \ln \left(\dfrac{x-a}{x+a}\right) + c \)
\( \dfrac{1}{a^2 - x^2} \) \( \dfrac{1}{2 a} \ln \left(\dfrac{a+x}{a-x}\right) + c \)

Rules and Properties of Indefinite Integrals

\( u \) and \( v \) are functions of \( x \)

Rules and Properties of Definite Integrals



More References and Links

Handbook of Mathematical Functions
Joel Hass, University of California, Davis; Maurice D. Weir Naval Postgraduate School; George B. Thomas, Jr.Massachusetts Institute of Technology ; University Calculus , Early Transcendentals, Third Edition , Boston Columbus , 2016, Pearson.
Gilbert Strang; MIT, Calculus, Wellesley-Cambridge Press, 1991 Engineering Mathematics with Examples and Solutions