How do you integrate $int sin(sqrtx)$ by integration by parts method?

Question

1558 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-09-02T17:19:26

$2sin(sqrt(x))-2 sqrt(x) cos(sqrt(x))+C$

Making $x = y^2$ in $int sin(sqrtx)dx$

after $dx = 2 y dy$ we have

$int sin(sqrtx)dx equiv 2inty sin y dy$

but $d/(dy)(y cos y) = cosy -y sin y$ so

$2inty sin y dy=2int cos y dy -2y cos y = 2sin y -2y cos y + C$

Finally

$int sin(sqrtx)dx = 2sin(sqrt(x))-2 sqrt(x) cos(sqrt(x))+C$

How do you integrate int sin(sqrtx)int sin(sqrtx) by integration by parts method?