# 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 $$u( - x) = - u(x)$$