Question

Simplify the integral.

Answer 1

\[\int\limits_{}^{}\frac{ 1 }{ 3 }[2xcos(x^2)-x^3]^2dx\]

Answer 2

\[\rightarrow 1/3\int\limits (4x^2\cos^2(x^2)-4xcos(x^2)+x^6) dx\]

Answer 3

If we expand..

Answer 4

I don't know how to take the integral of the cos^2 and cos^2(x^2)

Answer 5

Do you know the half angle formula and integration by parts?

Answer 6

no

Answer 7

Integration by parts tells us if you the integral \[\int\limits udv = uv - \int\limits vdu\]

Answer 8

@bmk614 have studied about Special integral S(u) C(u) and error functions?

Answer 9

I just do not know how to find the integral of cos^2(x^2)

Answer 10

Half angle formula:

\[\cos^2(x)=1/2-((\cos(2x))/2)\]

Answer 11

@daniel.ohearn1 the second term after completing the whole would be \[-4x^4\cos(x^2)\]

Answer 12

*completing the whole square

Answer 13

You will still need IBP and Half angle I believe.

Answer 14

What is
\[\int\limits_{}^{}4x^2\cos^2(x^2)\]

Answer 15

bmk should catch our errors too

Answer 16

use the formula I gave you. Or use the table method. Derive the first term and find the antiderivative the second term can you do that?

Answer 17

You will also need a special function. What is the integral of cos^2(x^2)?

Answer 18

I honestly have no clue.

Answer 19

Irjal might help you >

Answer 20

@ijlal

Answer 21

@bmk614 oh my oh my the integral is so damn lengthy first of all have you studied fresnel integral's as special integrals and error functions? if not then this integral my dear you typed it wrong maybe a typing mistake because without studying these special integrals you can never have a closed form solution and to check my answer i used this website http://www.integral-calculator.com/

Answer 22

and this is what i got and same as the online integral calculator

Answer 23

@daniel.ohearn1 i am a bit out of practice of these special integrals i might not be able to help you with this out of practice :/ @bmk614 do check again the question in your book did you type it wrong ?

Answer 24

It is not a question in a book. I know this is right.

Answer 25

use the website i gave you to check out the answer it shows the steps as well

Answer 26

check out the attachment above it contains the answer @bmk614

Answer 27

http://www.wolframalpha.com/input/?i=integrate+(4x%5E2)+cos%5E2(x%5E2)

Wolfram gets a similar answer.

Answer 28

The part that's difficult to see for me is how they made that substitution with the sine integral function.

Answer 29

@bmk614 any integral involving the angles which are squared or cubic examples are
all these contain special integrals i am afraid i wont be able to help you with these integrals as they involve the convergence of maclaurin series and etcc and it's been a long time i haven't studied them :/