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

1 Answer
Aug 25, 2016

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).