Question

Evaluate: $\int cosec^{3} {x} \ dx$

Answer 1

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We have:

I=∫csc3xdx

We will use integration by parts. First, rewrite the integral as:

I=∫csc2xcscxdx

Since integration by parts takes the form ∫udv=uv−∫vdu, let:

{u=cscx ⇒ du=−cotxcscxdxdv=csc2xdx ⇒ v=−cotx

Applying integration by parts:

I=−cotxcscx−∫cot2xcscxdx

Through the Pythagorean identity, write cot2xas csc2x−1.

I=−cotxcscx−∫(csc2x−1)(cscx)dx

I=−cotxcscx−∫csc3xdx+∫cscxdx

Note that I=∫csc3xdx and ∫cscxdx=−ln(|cotx+cscx|).

I=−cotxcscx−I−ln(|cotx+cscx|)

Add the original integral I to both sides.

2I=−cotxcscx−ln(|cotx+cscx|)

Solve for I and add the constant of integration:

I=−cotxcscx−ln(|cotx+cscx|)2+C

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