What is the integral of (csc^2(x)-1)^2 ?

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

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integral of (csc^2(x)-1)^2

\int (csc^{2} (x) - 1)^{2} d x

- \frac{1}{3} cot^{3} (x) + x + cot (x) + C

Solution steps

\int (csc^{2} (x) - 1)^{2} d x

Rewrite using trig identities

= \int cot^{4} (x) d x

Apply Integral Reduction

= - \frac{cot ^{3} ( x )}{3} - \int cot^{2} (x) d x

\int cot^{2} (x) d x = - x - cot (x)

= - \frac{cot ^{3} ( x )}{3} - (- x - cot (x))

Simplify

- \frac{cot ^{3} ( x )}{3} - (- x - cot (x)) : - \frac{1}{3} cot^{3} (x) + x + cot (x)

= - \frac{1}{3} cot^{3} (x) + x + cot (x)

Add a constant to the solution

= - \frac{1}{3} cot^{3} (x) + x + cot (x) + C

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