Question

How do you find the integral of `xe^x - sec(7x)tan(7x) dx`?

Answer 1

`int(xe^x-sec7xtan7x)dx=xe^x-e^x-1/7sec(7x)+C`

Explanation:

So, we want

`int(xe^x-sec7xtan7x)dx`. We can split up across the difference, yielding the following two integrals:

`intxe^xdx-intsec7xtan7xdx`

For `intxe^xdx`, we will use Integration by Parts, making the following selections:

`u=x`
`du=dx`
`dv=e^xdx`
`v=inte^xdx=e^x`

`uv-intvdu=xe^x-inte^xdx`

`=xe^x-e^x`

For `intsec7xtan7xdx`, let's first make a simple substitution to clean things up:

`u=7x`
`du=7dx`
`1/7du=dx`

Then, we have the common integral

`1/7intsecutanudu=1/7secu=1/7sec(7x)`

Combining our integrals together and putting in the constant of integration, we get

`int(xe^x-sec7xtan7x)dx=xe^x-e^x-1/7sec(7x)+C`