integral

An integral is an operation that calculates the continuous analog of a sum, useful for calculating areas and volumes.

A definite integral calculates the signed area of the region in the plane that is bounded by the graph of a given function between two points in the line.

Areas above the x-axis are considered positive and those below are considered negative
the integral of a functions is the positive signed area minus the negative signed area.

Some identities.

$\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+c,n\ne -1$

$\int e^{ax}dx=\frac{1}{a}{e}^{ax}+c$

$\int \frac{1}{x}dx=\ln (|x|)+c$

$\int \sin (ax)dx=-\frac{1}{a}\cos (ax)+c$

$\int \cos (ax)dx=\frac{1}{a}\sin (ax)+c$

$\int {\sec }^2(ax)dx=\frac{1}{a}\tan (ax)+c$

$\int {\csc }^2(ax)dx=-\frac{1}{a}\cot (ax)+c$

$\int \sec (ax)\tan (ax)dx=\frac{1}{a}\sec (ax)+c$

$\int \csc (ax)\cot (ax)dx=-\frac{1}{a}\csc (ax)+c$

$\int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}\left(\frac{x}{a}\right)+c,a>0$

$\int \frac{-1}{\sqrt{a^2-x^2}}dx={\cos }^{-1}\left(\frac{x}{a}\right)+c,a>0$

$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+c$

$\int (ax+b{)}^{n}dx=\frac{1}{a(n+1)}(ax+b{)}^{n+1}+c,n\ne -1$

$\int (ax+b{)}^{-1}dx=\frac{1}{a}\ln (|ax+b|)+c$