Question

How do you find `int xe^x - sec(7x)tan(7x)`?

Answer 1

I found: `xe^x-e^x-1/7*1/cos(7x)+c`

Explanation:

Let us write it as:
`intxe^xdx-intsec(x)tan(x)dx=`
we can use Integration by Parts for the first and a transformation for the second:
`=xe^x-int1e^xdx-int1/cos(7x)*sin(7x)/cos(7x)dx=`
`=xe^x-e^x-intsin(7x)/cos^2(7x)dx=`

we can consider that: `d[cos(7x)]=-7sin(7x)dx` and use this into the second integral and write (substituting and dividing by `7`):
`=xe^x-e^x+1/7int(d[cos(7x)])/cos^2(7x)=`
now we can integrate the second integral using `cos(7x)` as if it was our variable of integration and using the usual rules of integration to get:
`=xe^x-e^x-1/7*1/cos(7x)+c`