`int tan(x) sec^3(x) dx` Evaluate the integral

`int tan(x) sec^3(x)dx`

To solve, apply u-substitution method.

Let the u be:

`u = sec(x)`

Then, differentiate it.

`du = tan(x) sec(x) dx`

To be able to plug-in this to the integral, re-write the integrand as:

`int tan(x) sec^3(x) dx = sec^2(x) * tan(x) sec(x) dx`

Then, express the integrand in terms of u. So it becomes:

`=int u^2 du`

To take the integral of this, apply the formula `int u^n du = u^(n+1)/(n+1)+C` .

`= u^3/3 + C`

And, substitute back u= sec(x).

`= (sec^3(x))/3+C`

Therefore,  `int tan(x) sec^3(x) dx = (sec^3(x))/3+C` .