How can I solve this without using Reduction Formula?

[Dj Pedobear]

Homework Statement

integral of cot^2 x / csc^8 x dx

Homework Equations

u = cot x
du = csc^2 x du

The Attempt at a Solution

if I use reduction formula I could answer this but it's going to be very very LONG SOLUTION

I just need some basic integral work.

 

[Let'sthink]

First convert the expression to only sin and cos and then think of proper substitution. If youdo not wish to use reduction formula you need to juggle up the expressions and remove powers to get expression in terms of multiple angles.

 

[Ray Vickson]

First convert the expression to only sin and cos and then think of proper substitution. If youdo not wish to use reduction formula you need to juggle up the expressions and remove powers to get expression in terms of multiple angles.

Homework Statement

integral of cot^2 x / csc^8 x dx

Homework Equations

u = cot x
du = csc^2 x du

The Attempt at a Solution

if I use reduction formula I could answer this but it's going to be very very LONG SOLUTION

I just need some basic integral work.

Write the integrand as $f(x) = \cos^2(x) \sin^6 (x)$. Use $\cos^2(x) = 1 -\sin^2(x)$ to get your integral $F = \int f(x) \, dx$ in the form $F = I_6-I_8$, where $I_n = \int \sin^n(x) \, dx$.

Apply integration by parts to $I_n$, using $u = \sin^{n-1}(x)$ and $dv = \sin(x) \, dx$. This gives
$$I_n = -\cos(x) \sin^{n-1}(x) + (n-1) \int \cos^2(x) \sin^{n-2}(x) \, dx = -\cos(x) \sin^{n-1}(x) + (n-1) [I_{n-2} - I_n]$$
This is an equation connecting $I_n$ to $I_{n-2}$, so you can solve it to express $I_n$ in terms of $\sin(x), \cos(x)$ and $I_{n-2}$. Finally, you can express $I_8$ in terms of $I_6$, then $I_6$ in terms of $I_4$, etc. The answer you want will drop out pretty quickly and easily.

 

[Dj Pedobear]

thx :D