Question

What is `f(x) = int 2sinx-xcosx dx` if `f((7pi)/6) = 0`?

Answer 1

`f(x)=-3cosx-xsinx-(3sqrt3)/2-(7pi)/12`

Explanation:

Split up the integral into
`f(x)=2intsinxdx-intxcosxdx`
Using integration by parts, we get
`f(x)=-2cosx-xsinx-cosx+C`
`f(x)=-3cosx-xsinx+C`
Substituting `x=(7pi)/6`, we get
`f((7pi)/6)=0=-3cos((7pi)/6)-(7pi)/6sin((7pi)/6)+C`
Solving for C, we get
`C=-(3sqrt3)/2-(7pi)/12`
Therefore, `f(x)=-3cosx-xsinx-(3sqrt3)/2-(7pi)/12`.