What is the integral of (sec^2(x))/(tan^3(x)) ?

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

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integral of (sec^2(x))/(tan^3(x))

\int \frac{sec ^{2} ( x )}{tan ^{3} ( x )} d x

- \frac{1}{2} cot^{2} (x) + C

Solution steps

\int \frac{sec ^{2} ( x )}{tan ^{3} ( x )} d x

Apply u-substitution

= \int \frac{1}{u ^{3}} d u

Apply the Power Rule

= - \frac{1}{2 u ^{2}}

Substitute back

u = tan (x)

= - \frac{1}{2 tan ^{2} ( x )}

Simplify

- \frac{1}{2 tan ^{2} ( x )} : - \frac{1}{2} cot^{2} (x)

= - \frac{1}{2} cot^{2} (x)

Add a constant to the solution

= - \frac{1}{2} cot^{2} (x) + C

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