Question

What is `f(x) = int xcosx dx` if `f(pi/4)=-2`?

Answer 1

`f(x)=xsinx+cosx-2-1/sqrt2(pi/4+1)`.

Explanation:

`f(x)=intxcosxdx`

We have to integrate by using, the Rule of Integration by Parts :

`intuvdx=uintvdx-int((du)/dxintvdx)dx`

We take `u=x rArr (du)/dx=1`, and,

`v=cosx rArr intvdx=sinx`

Hence, `f(x)=xsinx-intsinxdx=xsinx+cosx+C`

To determine the const. of Integration `C`, we use,

`f(pi/4)=-2 rArr pi/4sin(pi/4)+cos(pi/4)+C=-2`

`rArr pi/4*1/sqrt2+1/sqrt2+C=-2`

`rArr C=-2-1/sqrt2(pi/4+1)`

Therefore, `f(x)=xsinx+cosx-2-1/sqrt2(pi/4+1)`.