Asked by mat

Find the integral.

∫ -10 x sin x^2 dx

Answers

Answered by Reiny
wouldn't that just be
5 cos(x^2) + c ?
Answered by mat
Yes, but how would i show the work.
Answered by Reiny
some patterns you should just recognize.

I know how to differentiate and integrate sin (?),
but in both cases the derivative of the "angle" which would be the x from the x^2 has to be considered.

Sure enough, the derivative of x^2, which would be 2x , is hanging around at the front as a multiple
( 5(2x) = 10x)

Formal way:

let x^2 = u
2x = du/dx
dx = du/(2x)

∫ -10 x sin x^2 dx
= ∫ -10 x sin (u) (du/(2x))
= ∫ -5 sin (u) du
= 5 cos(u) + c
= 5 cos(x^2) + c
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