Question

Evaluate the Intergal

Answer 1

\[\int\limits_{}^{}x^2\cos(2x) dx\]

Answer 2

WITHOUT integration by parts.

Answer 3

I am only allowed to use u-substitution but I can't see how to manipulate it so it works :/ .

Answer 4

doing it without integration by parts eh? noooo idea

Answer 5

Thanks anyways :P .

Answer 6

maybe something like? \[\int\limits_{}^{}u*\cos{du}\]

Answer 7

Maybe. But that 2x is INSIDE the cosine so that won't work.

Answer 8

cos(2x) = cos^2(x) - sin^2(x)

Answer 9

don't know if that will help...

Answer 10

@Loser66 \(du=2\cos 2x\,dx\)

Answer 11

@Loser66 wouldn't that reduce to \(\int\frac12\arcsin^2u\,du\)... ?

Answer 12

well, the following technique makes no use of parts... but I'm not sure if series are allowed :-)
$$\cos x=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n}\\\cos(2x)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}(2x)^{2n}\\x^2\cos(2x)=\sum_{n=0}^\infty\frac{(-1)^n4^n}{(2n)!}x^{2(n+1)}\\\int x^2\cos(2x)\,dx=\sum_{n=0}^\infty\frac{(-1)^n4^n}{(2(n+1))!}x^{2(n+1)+1}$$that form suggests it can be broken down into two power series of trigonometric functions

Answer 13

I don't think it is allowed when it's not allowed using by parts. This method is taught in the end of Cal2 course.

Answer 14

right @Loser66 you have a good point...

Answer 15

ty, but it doesn't help the Asker. :(

Answer 16

nvm... I've got it. I think you may be able to use a Weierstrass substitution.

Answer 17

No it's okay. This was for my friend and she just said to use integration by parts :P .

Answer 18

well here's a wiki page just in case :-)
http://en.wikipedia.org/wiki/Weierstrass_substitution

Answer 19

There is that :P .