What is the integral of (1+cot^2(θ))(csc^2(θ)) ?

The integral of (1+cot^2(θ))(csc^2(θ)) is -1/3 csc^3(θ)cos(θ)-2/3 cot(θ)+C

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integral of (1+cot^2(θ))(csc^2(θ))

\int (1 + cot^{2} (θ)) (csc^{2} (θ)) d θ

- \frac{1}{3} csc^{3} (θ) cos (θ) - \frac{2}{3} cot (θ) + C

Solution steps

\int (1 + cot^{2} (θ)) csc^{2} (θ) d θ

Rewrite using trig identities

= \int csc^{4} (θ) d θ

Apply Integral Reduction

= - \frac{cos ( θ ) csc ^{3} ( θ )}{3} + \frac{2}{3} \cdot \int csc^{2} (θ) d θ

\int csc^{2} (θ) d θ = - cot (θ)

= - \frac{cos ( θ ) csc ^{3} ( θ )}{3} + \frac{2}{3} (- cot (θ))

Simplify

- \frac{cos ( θ ) csc ^{3} ( θ )}{3} + \frac{2}{3} (- cot (θ)) : - \frac{1}{3} csc^{3} (θ) cos (θ) - \frac{2}{3} cot (θ)

= - \frac{1}{3} csc^{3} (θ) cos (θ) - \frac{2}{3} cot (θ)

Add a constant to the solution

= - \frac{1}{3} csc^{3} (θ) cos (θ) - \frac{2}{3} cot (θ) + C

What is the integral of (1+cot^2(θ))(csc^2(θ)) ?
The integral of (1+cot^2(θ))(csc^2(θ)) is -1/3 csc^3(θ)cos(θ)-2/3 cot(θ)+C