1. Integration of 1/sin^4(x)

I need help integrating this function. I've done integration by parts many times, and it seems to be an infinite loop. Please help, would be greatly appreciated!

2. Re: Integration of 1/sin^4(x)

Originally Posted by mathimaticool
I need help integrating this function. I've done integration by parts many times, and it seems to be an infinite loop. Please help, would be greatly appreciated!
$\frac{1}{\sin^4(x)}=\csc^4(x)=\csc^2(x)\csc^2(x)=( 1+\cot^2(x))\csc^2(x)$

Now let $u=\cot(x) \implies du=-\csc^2(x)dx$

Can you finish from here?

3. Re: Integration of 1/sin^4(x)

Ahh ok. But when I get to

-cot^2(x) - integral(csc^2(x)cot(x) dx)

I don't know what to do. Thanks a lot

4. Re: Integration of 1/sin^4(x)

Originally Posted by mathimaticool
Ahh ok. But when I get to

-cot^2(x) - integral(csc^2(x)cot(x) dx)

I don't know what to do. Thanks a lot
So you have

$\int (1+\cot^2(x))\csc^2(x) dx$

If you let $u=\cot(x) \implies du = -\csc^2(x)$

This gives

$\int (1+\cot^2(x))\csc^2(x) dx=\int (1+u^2)(-du)=-\int (1+u^2)du$

5. Re: Integration of 1/sin^4(x)

AHHH thanks soo much! Got it!

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integral of csc^4x

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