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

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

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

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

- \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Apply Integral Reduction

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

\int cot (x) d x = ln ∣ sin (x) ∣

= - \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣

Add a constant to the solution

= - \frac{cot ^{2} ( x )}{2} - ln ∣ sin (x) ∣ + C

\frac{d}{d x} (\frac{1}{6 x - 7})

x \to - 1 lim (3 x^{2} - 3 x + 3)

\frac{d}{d x} ((- 1)^{n} x^{n})

\frac{d}{d x} (\frac{1 + x}{x})

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