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Integrate Sine and Square root Composite Function

Petrus

Well-known member
Feb 21, 2013
739
Hello MHB,
I got stuck on integrate this function

\(\displaystyle \int \frac{\sin^3(\sqrt{x})}{\sqrt{x}}dx\)
my first thinking was rewrite it as \(\displaystyle \int \frac{\sin^2(\sqrt{x})\sin(\sqrt{x})}{\sqrt{x}}dx\)
then use the identity \(\displaystyle \cos^2(x)+\sin^2(x)=1 \ \therefore \sin^2x=1- \cos^2(x)\)
\(\displaystyle \int \frac{(1-\cos^2(\sqrt{x}))\sin(\sqrt{x})}{\sqrt{x}}dx\)
subsitute \(\displaystyle u= \cos(x) \therefore du=- \sin(x) dx\)
then we get
\(\displaystyle - \int \frac{1-u^2}{ \cos^{-1}(u)}du\)
but that does not seem smart so my last ide is integrate by part, but I struggle on that part..

\(\displaystyle u= \sqrt{x}^{-1} \therefore du=\sqrt{x}^{-2}\) and \(\displaystyle dv=\sin^3(\sqrt{x}) \therefore v=?\)

Regards,
\(\displaystyle |\pi\rangle\)
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Re: Integrate 2

Use the substitution $\sqrt{x}=u$
 

Petrus

Well-known member
Feb 21, 2013
739
Re: Integrate 2

Use the substitution $\sqrt{x}=u$
Hello Zaid,
I don't see what is the point with that=S? can I substitute twice or ? I am kinda clueless:confused:

Regards,
\(\displaystyle |\pi\rangle\)
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
Re: Integrate 2

Hello Zaid,
I don't see what is the point with that=S? can I substitute twice or ? I am kinda clueless:confused:

Regards,
\(\displaystyle |\pi\rangle\)
One thing to realize that \(\displaystyle (\sqrt{x})'=\frac{1}{2\sqrt{x}}\)

So before we proceed , we make \(\displaystyle u=\sqrt{x}\) substitution , which makes things easier since we are left with

\(\displaystyle 2\int \sin^3 (u)\,du \) which you can integrate , right ?

I have got to get some sleep now , if you are still stuck someone is always around :cool:.
 

Petrus

Well-known member
Feb 21, 2013
739
Re: Integrate 2

One thing to realize that \(\displaystyle (\sqrt{x})'=\frac{1}{2\sqrt{x}}\)

So before we proceed , we make \(\displaystyle u=\sqrt{x}\) substitution , which makes things easier since we are left with

\(\displaystyle 2\int \sin^3 (u)\,du \) which you can integrate , right ?

I have got to get some sleep now , if you are still stuck someone is always around :cool:.
Hello Zaid,
Now it make alot sense!:) Thanks for taking your time and sleep well! I am also supposed to sleep but will do it soom =D

Regards,
\(\displaystyle |\pi\rangle\)