1. ## long trigo integral:

$\displaystyle \int {\csc^5{x}}dx$

Solutions:
$\displaystyle \int {\csc^3{x}}(1+\cot^2{x})dx$

$\displaystyle \int {\csc^3{x}}dx + \int {\cot^2{x}}{\csc^3{x}}dx$

solving for int of $\displaystyle \int {\csc^3{x}}dx$
$\displaystyle \frac{1}{2}(-\cot{x}\csc{x} + \ln |\csc{x} - \cot{x}|)$

this is where i got stuck for solving for $\displaystyle \int {\cot^2{x}}{\csc^3{x}}dx$ :

$\displaystyle \int {\cot^2{x}}{\csc^3{x}}dx$

$\displaystyle \int {\csc^5{x} - \csc^3{x} }dx$
which is repeating the csc^5{x}

help with hints

thanks

2. $\displaystyle \int \cot^{2}x \ \csc^{3}x \ dx =$$\displaystyle \int (\cot^{2}x )(1+\cot^{2}x) \csc x \ dx$

So $\displaystyle \int \cot^{2}x \csc x \ dx + \int \cot^{4}x \ \csc x \ dx$

3. now confused with the integral of
$\displaystyle \int \cot^{4}x \ \csc x \ dx$

now confused with the integral of
$\displaystyle \int \cot^{4}x \ \csc x \ dx$

I don't know about anyone else, but I typically convert everything to sine and cosine functions:
$\displaystyle \int cot^{4}(x)csc(x)dx = \int \frac{cos^4(x)}{sin^5(x)} dx$

Do this by parts:
$\displaystyle u = cos(x) \implies du = sin(x) dx$

$\displaystyle dv = \frac{cos^3(x)}{sin^5(x)} dx$

We need v.
$\displaystyle v = \int \frac{cos^3(x)}{sin^5(x)} dx$

$\displaystyle v = \int \frac{cos^2(x)}{sin^5(x)} cos(x) dx$

Let $\displaystyle z = sin(x) \implies dz = cos(x) dx$
$\displaystyle v = \int \frac{cos^2(x)}{sin^5(x)} cos(x) dx = \int \frac{1 - z^2}{z^5} dz = \int \frac{1}{z^5} dz - \int \frac{1}{z^3}$

$\displaystyle v = -\frac{1}{4z^4} - \frac{1}{2z^2} = - \left ( \frac{1}{4sin^4(x)} + \frac{1}{2sin^2(x)} \right )$

So back to the original integral:
$\displaystyle \int cot^{4}(x)csc(x)dx = \int \frac{cos^4(x)}{sin^5(x)} dx$

$\displaystyle = - cos(x) \cdot \left ( \frac{1}{4sin^4(x)} + \frac{1}{2sin^2(x)} \right ) + \int \left ( \frac{1}{4sin^4(x)} + \frac{1}{2sin^2(x)} \right ) \cdot sin(x) dx$

Can you go from here?

-Dan

$\displaystyle \int {\csc^5{x}}dx$