Question

What is the integrate of Csc^5x

Answer 1

The original equation:

Integral of csc(5x)

Use a u sub:

u = 5x

du = 5dx

Simplify the du: $\frac{1}{5} du = dx$

Apply to the equation:

Integral csc(u)$\frac{1}{5}$du

Simplify:

Integral $\frac{1}{5}$$\frac{1}{sin(u)}$du

Factor out the constant:

$\frac{1}{5}$ Integral $\frac{1}{sin(u)}$du

Use a second Substitution:

v = tan($\frac{u}{2}$)

du = $\frac{2}{1+v^{2} }$dv

Applying to the equation:

$\frac{1}{5} Integral \frac{1}{ \frac{2v}{1+ v^{2} } } * \frac{2}{1+v^{2} } dv$

Simplify:

$\frac{1}{5} Integral \frac{1}{v} dv$

Integrate:

$\frac{1}{5} ln(|v|)$

Insert back in your v and u:

v = tan($\frac{u}{2}$)

u = 5x

This gives us the final equation (don't forget your constant):

$\frac{1}{5} ln(|tan( \frac{5x}{2} )|) + C$

(Thank you for making me write it out, I made a mistake on the original answer.)