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#