How Do You Solve the Integral of t * csc^2(t) dt?

[mcmw]

Homework Statement

integral (t * csc^2 (t) ) dt

Homework Equations

The Attempt at a Solution

t * -cot (t) - integral( 1 * -cot (t)) u= t dv= csc^2(t)
du= 1 v= - cot (t)
-t * cot(t) - ? I don't understand how to find the antiderivative of -cot(t)

 

[G01]

Integration by parts is the way I'd also go about this.

Hint: In terms of other trig functions, what is cotangent equal to? You should end up with something that is solvable by substitution.

 

[christianjb]

The integral contains singularities whenever sin(x)=0, or x=n PI

If it's an integral over -T, to T then the integral is zero (by symmetry)

 

[christianjb]

Well, the Integrator gives an answer.
http://integrals.wolfram.com/index.jsp

The integral of cot(x)=log(sin(x))

(yes, I know I'm cheating!)

 

[Gib Z]

\cot x = \frac{\cos x}{\sin x}

\int \cot x dx = \int \frac{\cos x}{\sin x} dx.

let u= sin x, then du = cos x dx

\int \frac{1}{u} du = \ln u + C = \ln (\sin x) + C