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## Basic Integrals

· Constant factor Rule

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\int cxdx=c\int xdx$
· Sum Rule of Integration

The integral of the sum of two or more functions is equal to the sum of their integrals

$\int\left(a+b\right)dx=\int adx+\int bdx$

## Trigonometric integrals

$\int\sec\left(x\right)\tan\left(x\right)^2dx=\int\sec\left(x\right)^3dx-\int\sec\left(x\right)dx$

The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$

$\int\sec\left(x\right)dx=\ln\left|\sec\left(x\right)+\tan\left(x\right)\right|+C$

Simplify the integral $[root]$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$

$\int\sec\left(x\right)^ndx=\frac{\sin\left(x\right)\sec\left(x\right)^{\left(n-1\right)}}{n-1}+\frac{n-2}{n-1}\int\sec\left(x\right)^{\left(n-2\right)}dx$

## Basic Derivatives

· Derivative of the linear function

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(x\right)=1$

## Derivatives of trigonometric functions

Taking the derivative of secant

$\frac{d}{dx}\left(\sec\left(x\right)\right)=\sec\left(x\right)\tan\left(x\right)\frac{d}{dx}\left(x\right)$
$\int\frac{x^2}{\sqrt{x^2+6}}dx$

### Main topic:

Integrals of Rational Functions

### Time to solve it:

~ 0.21 s (SnapXam)