Question

What is `f(x) = int x-sin2x-6cosx dx` if `f(pi/2)=3`?

Answer 1

`f(x)=x^2/2+1/2cos2x-6sinx+7/2-pi^2/8`

Explanation:

`f(x)=int(x-sin2x-6cosx)dx` `" "f(pi/2)=3`

integrate term by term

`f(x)=x^2/2+1/2cos2x-6sinx+C`

`f(pi/2)=1/2xx(pi/2)^2+1/2cos(2xxpi/2)-6sin(2xxpi/2)+C=3`

`pi^2/8+1/2cancel(cospi)^-1- 6cancel(sinpi)^0+C=3`

`pi^2/8-1/2+C=3`

`C=3+1/2-pi^2/8=7/2-pi^2/8`

`f(x)=x^2/2+1/2cos2x-6sinx+7/2-pi^2/8`