# Evaluate the integral by making an appropriate substitution. \int \tan x \sec^{4} x dx

## Question:

Evaluate the integral by making an appropriate substitution.

{eq}\displaystyle\; \int\tan x \sec^{4} x\,dx {/eq}

## Indefinite Integrals :

Solving that indefinite integral where the substitution of the variable is to be made, then the derivative of the variable is applied and then the integral is replaced with the new indefinite integral.

We have the indefinite integral given as;

{eq}\int\tan x \sec^{4} x\,dx\\ {/eq}

Now, if we take the substitution of:

{eq}u=\sec \left(x\right)\\ {/eq}

we have differentiation as:

{eq}\frac{du}{dx}=\sec \left(x\right)\tan \left(x\right)\\ {/eq}

So the integral will be:

{eq}\int\tan x \sec^{4} x\,dx\\=\int \:u^3du\\ =\frac{u^{3+1}}{3+1}+c\\ =\frac{1}{4}\sec ^4\left(x\right)+C {/eq}