needing to integrate (cos(a sin(a)))^6

[Knight]

kk lol This is my last one... Please help me intergrate this!

(cos(asin(a)))^6

 

[soroban]

Re: Help

Hello, Knight!

You don't seem to have a variable.
I'll take a guess at what you meant . . .

\(\displaystyle \L\int\)$\displaystyle \cos^6(\arcsin x)\,dx$

Let $\displaystyle \theta \:=\:\arcsin x\;\;\Rightarrow\;\;\sin\theta \:=\:x$

Then $\displaystyle \theta$ is an angle in a right triangle
. . with: $\displaystyle opp\,=\,x,\;hyp\,=\,1$
Hence: $\displaystyle \,adj\,=\,\sqrt{1\,-\,x^2}\,$ and $\displaystyle \,\cos\theta \,=\,\sqrt{1\,-\,x^2}$

So we have: \(\displaystyle \L\:\int\)\(\displaystyle cos^6\theta\,d\theta \;=\;\L\int\)\(\displaystyle \left(\sqrt{1\,-\,x^2}\right)^6\,dx \;=\;\L\int\)$\displaystyle (1\,-\,x^2)^3\,dx$

. . \(\displaystyle = \;\L\int\)$\displaystyle (1\,-\,3x^2\,+\,3x^4\,-\,x^6)\,dx$

And you can finish it now . . .

 

[Knight]

No, thats not it lol

The variable was (a). I guess that could of been confusing so replace it with x.
Please help!

 

[stapel]

Please help!

What is lacking in the help you already received?

Please be specific. Thank you.

Eliz.