What is $f (x) = \int x sin x d x$ $f(x) = int xsinx dx$ if $f (\frac{π}{4}) = - 2$ $f(pi/4)=-2$ ?

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Answer 1 · 2016-06-29T09:24:33

$= - x cos x + sin x + \frac{1}{\sqrt{2}} (\frac{π}{4} - 1) - 2$ $= - x cos x + sin x + 1/sqrt 2 (pi/4 -1) - 2$

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

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

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

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

What is f(x) = int xsinx dxf(x)=∫xsinxdxf(x) = int xsinx dx if f(pi/4)=-2 f(π4)=−2f(pi/4)=-2 ?