# How do I work out #intcot(x)dx# by substitution?

##### 1 Answer

The key to this question is to use your trig identities to rewrite the integral:

#int cot(x) dx = int cos(x) / sin(x) dx#

Now remember that what we are trying to find with substitution is some term being multiplied by it's derivative, which allows us to make use of a variant of the Chain Rule for derivatives that lets us work backwards:

#F'(x) = f'(g(x))g'(x)#

#F(x) = int f'(g(x))g'(x)#

#F(x) = int f'(u)du # where#u = g(x)#

Setting

Setting

Rewriting our integral:

#int cos(x) / sin(x) dx# =>#int (du)/u => int u^(-1)du# , where#u = sin(x)#

We must now remember that the Power Rule doesn't apply when the power is

#int u^(-1)du = ln(u) + C#

Plugging our

#int cot(x) dx = ln( sin(x) ) + C#