Question

integrate the following: `inte^x[tanx+sec^2x]` ?

Answer 1

`I=e^xtan(x)+C`

Explanation:

We want to solve

`I=inte^x(tan(x)+sec^2(x))dx`

Split into two integrals

`I=inte^xtan(x)dx+inte^xsec^2(x)dx`

Use integration by parts (first integral)

`intudv=uv-intvdu`

Let `u=tan(x)=>du=sec^2(x)dx`

And `dv=e^xdx=>v=e^x`

`I=e^xtan(x)-inte^xsec^2(x)dx+inte^xsec^2(x)dx`

A fortunate cancellation

`I=e^xtan(x)+C`

Answer 2

`e^x*tanx+c`

Explanation:

`I=inte^x[tanx+sec^2x]dx`
`color(red)(I=inte^x[f(x)+f^'(x)]dx=e^xf(x)+c`
Here, `f(x)=tanx=>f^'(x)=sec^2x`
So, `I=inte^x[tanx+sec^2x]dx=e^x*tanx+c`