Antiderivative Rules

See also Derivative Rules.

Basic Functions

Let $a$ and $n$ be arbitrary constants.

$$\begin{array}{rlrlr}& \int adx& & =ax+C& \\ & \int x^{n}dx& & =\frac{1}{n+1}{x}^{n+1}+C& \text{(power rule, for n=−1)}\end{array}$$

The power rule cannot be used when $n=-1$, since it would require division by zero. That case, $\int \frac{1}{x}dx$, is covered in the next section.

Exponential and Logarithmic Functions

Let $b$ be an arbitrary positive constant.

$$\begin{array}{rlrlr}& \int e^{x}dx& & =e^x+C& \\ & \int b^{x}dx& & =\frac{1}{\ln b}{b}^x+C& \\ & \int \frac{1}{x}dx& & =\ln |x|+C& \text{for }x\ne 0\end{array}$$

Trigonometric Functions

$$\begin{array}{rlrl}& \int \sin xdx& & =-\cos x+C\\ & \int \cos xdx& & =\sin x+C\\ & \int \tan xdx& & =-\ln |\cos x|+C\\ & \int \cot xdx& & =\ln |\sin x|+C\\ & \int \sec xdx& & =\ln |\sec x+\tan x|+C\\ & \int \csc xdx& & =-\ln |\csc x+\cot x|+C\\ & \int {\sec }^{2}xdx& & =\tan x+C\\ & \int {\csc }^{2}xdx& & =-\cot x+C\\ & \int \sec x\tan xdx& & =\sec x+C\\ & \int \csc x\cot xdx& & =-\csc x+C\end{array}$$

Inverse Trigonometric Functions

$$\begin{array}{rlrl}& \int \frac{1}{\sqrt{1-x^2}}dx& & =\arcsin x+C\\ & \int \frac{1}{x^2+1}dx& & =\arctan x+C\\ & \int \frac{1}{x\sqrt{x^2-1}}dx& & =\{\begin{array}{ll}\text{arcsec }x+C& \text{if }x>0\\ -\text{arcsec }x+C& \text{if }x<0\end{array}\end{array}$$

Hyperbolic Trigonometric Functions

Most calculus courses, including AP Calculus, do not cover these functions. You can ignore these if they are outside the scope of your class.

$$\begin{array}{rlrl}& \int \sinh xdx& & =\cosh x+C\\ & \int \cosh xdx& & =\sinh x+C\\ & \int \tanh xdx& & =\ln |\cosh x|+C\\ & \int \text{coth }xdx& & =\ln |\sinh x|+C\\ & \int {\text{sech}}^{2}xdx& & =\tanh x+C\\ & \int {\text{csch}}^{2}xdx& & =-\text{coth }x+C\\ & \int \text{sech }x\tanh xdx& & =-\text{sech }x+C\\ & \int \text{csch }x\text{coth }xdx& & =-\text{csch }x+C\end{array}$$

Inverse Hyperbolic Trigonometric Functions

Most calculus courses, including AP Calculus, do not cover these functions. You can ignore these if they are outside the scope of your class.

$$\begin{array}{rlrl}& \int \frac{1}{\sqrt{x^2+1}}dx& & =\text{arcsinh }x+C\\ & \int \frac{1}{\sqrt{x^2-1}}dx& & =\text{arccosh }x+C\\ & \int \frac{1}{1-x^2}dx& & =\text{arctanh }x+C\\ & \int \frac{1}{x\sqrt{1-x^2}}dx& & =-\text{arcsech }x+C\\ & \int \frac{1}{x\sqrt{1+x^2}}dx& & =\{\begin{array}{ll}-\text{arccsch }x+C& \text{if }x>0\\ \text{arccsch }x+C& \text{if }x<0\end{array}\end{array}$$

General Properties and Function Combinations

Let $f$ and $g$ be arbitrary functions and let $a$ and $b$ be arbitrary constants.

$$\begin{array}{rlrlr}& \int af(x)\pm bg(x)dx& & =a\int f(x)dx\pm b\int g(x)dx& \text{(linearity)}\\ & \int f^{\prime }(g(x))g^{\prime }(x)dx& & =f(g(x))+C& \text{(u-sub.)}\\ & \int f^{\prime }(x)g(x)dx& & =f(x)g(x)-\int f(x)g^{\prime }(x)dx& \text{(int. by parts)}\end{array}$$

If $f$ is an even function and $a$ is an arbitrary constant, then

$${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx.$$

If $f$ is an odd function and $a$ is an arbitrary constant, then

$${\int }_{-a}^{a}f(x)dx=0.$$

For any function $f$ and arbitrary constants $a$, $b$, and $c$, we have

$${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx.$$

Fundamental Theorem of Calculus

Version 1

If $F^{\prime }(x)=f(x)$ and if $a$ and $b$ are arbitrary constants, then

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

Version 2

Let $f(x)$ be a function and $a$ be an arbitrary constant.

$$\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)$$