What is f(x) = int xsinx dxf(x)=xsinxdx if f(pi/4)=-2 f(π4)=2?

1 Answer
Jun 29, 2016

= - x cos x + sin x + 1/sqrt 2 (pi/4 -1) - 2=xcosx+sinx+12(π41)2

Explanation:

f(x) = int dx \ (x \ sinx)

just IBP its is easiest way
Wikipedia

Here
u = x , u' =1

v' = sin x, v = - cos x

So we have

- x cos x - int dx \ (- cos x * 1)

= - x cos x + int dx \ ( cos x)

= - x cos x + sin x + C

now using the IV

-2 = - pi/4 1/sqrt 2 + 1/sqrt 2 + C

C = 1/sqrt 2 (pi/4 -1) -2