Formulas and Rules for Integrals in Calculus

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

Formulas

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

Let \( u \) and \( v \) be functions of \( x \)

Linearity: \( \displaystyle \int (a u + b v) dx = a \int u dx + b \int v dx \) , where \( a \) and \( b \) are constants.
Integration by parts: \( \displaystyle \int u dv = u v - \int v du \)
Derivative of an indefinite integral: \( \displaystyle \dfrac{d}{dx} \int u dx = u \)
Antiderivative: If \( \displaystyle \int u dx = U(x) + c \), then \( U(x) + c \) is called the antiderivative and \( U'(x) = u(x) \)
Integration of the quotient of the derivative and the function: \( \displaystyle \int \dfrac{u'}{u} dx = \ln | u | + c \)
Integration of a linearly composed function: \( \displaystyle \int u(a x+b) dx = \dfrac{1}{a} U(ax + b) + c \) , where \( \displaystyle U(x) = \int u dx \) and \( a \) and \( b \) are constants.
Integration of the product of the derivative and the function: \( \displaystyle \int u u' dx = \dfrac{1}{2} u^2 + c \)

Definite Integral from Indefinite Integral: If \( \displaystyle \int u(x) dx = U(x) + c \) then \( \displaystyle \int_{x_1}^{x_2} u(x) dx = ( U(x_2) + c) - ( U(x_1) +c ) = (U(x) +c) \Large{\left.\right|_{x_1}^{x_2}} \)
Changing the Order of Limits of Integration: \( \displaystyle \int_{x_1}^{x_2} u(x) dx = - \int_{x_2}^{x_1} u(x) dx \)
Equal Limits of Integration: \( \displaystyle \int_{x_1}^{x_1} u(x) dx = 0 \)
Splitting an Interval of Integration: \( \displaystyle \int_{x_1}^{x_3} u(x) dx = \int_{x_1}^{x_2} u(x) dx + \int_{x_2}^{x_3} u(x) dx \)
If \( u(x) \ge v(x) \) on the interval \( [x_1 , x_2] \), then \( \displaystyle \int_{x_1}^{x_2} u(x) dx \ge \displaystyle \int_{x_1}^{x_2} v(x) dx \)
Special case of the above: If \( u(x) \ge 0 \) on the interval \( [x_1 , x_2] \), then \( \displaystyle \int_{x_1}^{x_2} u(x) dx \ge 0 \)
If \( w(x) \le u(x) \le v(x) \) on the interval \( [x_1 , x_2] \), then \( \displaystyle \int_{x_1}^{x_2} w(x) dx \le \int_{x_1}^{x_2} u(x) dx \le \displaystyle \int_{x_1}^{x_2} v(x) dx \)
Even Function: \( \displaystyle \int_{-x_1}^{x_1} u(x) dx = 2 \displaystyle \int_{0}^{x_1} u(x) dx \) , if \( u(x) \) is an even function meaning \( u(-x) = u(x) \)
Odd Function: \( \displaystyle \int_{-x_1}^{x_1} u(x) dx = 0 \) , if \( u(x) \) is an odd function meaning \( u(-x) = -u(x) \)