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

2 Answers
Mar 11, 2018

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

Mar 11, 2018

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