# Formulas and Rules for Integrals in Calculus

 In what follows, $c$ is the constant of integration.

## Formulas

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

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

## Rules and Properties of Indefinite Integrals

$u$ and $v$ are functions of $x$
• Linearity: $\displaystyle \int (a u + b v) dx = a \int u dx + b \int v dx$ , $a$ 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 $\displaystyle \int u dx = U(x) + c$ , $U(x) + c$ is called the antiderivative and $U'(x) = u(x)$
• Integration of ratio of derivative and function: $\displaystyle \int \dfrac{u'}{u} dx = ln | u | + c$
• Integration of linearly composite 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 product of derivative and function: $\displaystyle \int u u' dx = \dfrac{1}{2} u^2 + c$

## Rules and Properties of Definite Integrals

• Definite Integral from Indefinite Integral : If $\displaystyle \int u(x) dx = U(x) + c$ then $\displaystyle \int_{x_1}^{x_2} u(x) dx = ( U(x_2) + c) - ( U(x_1) +c ) = (U(x) +c) \Large{\left.\right|_{x_1}^{x_2}}$
• Change Order of the Limits of Integration : $\displaystyle \int_{x_1}^{x_2} u(x) dx = - \int_{x_2}^{x_1} u(x) dx$
• Limits of Integration Equal : $\displaystyle \int_{x_1}^{x_1} u(x) dx = 0$
• Split 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 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 $u( - x) = - u(x)$