What is the integral of (sec^2(x)+sec^5(x))/(cot(x)) ?

The integral of (sec^2(x)+sec^5(x))/(cot(x)) is (tan^2(x))/2+(sec^5(x))/5+C

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integral of (sec^2(x)+sec^5(x))/(cot(x))

\int \frac{sec ^{2} ( x ) + sec ^{5} ( x )}{cot ( x )} d x

\frac{tan ^{2} ( x )}{2} + \frac{sec ^{5} ( x )}{5} + C

Solution steps

\int \frac{sec ^{2} ( x ) + sec ^{5} ( x )}{cot ( x )} d x

Expand

\frac{sec ^{2} ( x ) + sec ^{5} ( x )}{cot ( x )} : \frac{sec ^{2} ( x )}{cot ( x )} + \frac{sec ^{5} ( x )}{cot ( x )}

= \int \frac{sec ^{2} ( x )}{cot ( x )} + \frac{sec ^{5} ( x )}{cot ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{sec ^{2} ( x )}{cot ( x )} d x + \int \frac{sec ^{5} ( x )}{cot ( x )} d x

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

\int \frac{sec ^{5} ( x )}{cot ( x )} d x = \frac{sec ^{5} ( x )}{5}

= \frac{tan ^{2} ( x )}{2} + \frac{sec ^{5} ( x )}{5}

Add a constant to the solution

= \frac{tan ^{2} ( x )}{2} + \frac{sec ^{5} ( x )}{5} + C

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What is the integral of (sec^2(x)+sec^5(x))/(cot(x)) ?
The integral of (sec^2(x)+sec^5(x))/(cot(x)) is (tan^2(x))/2+(sec^5(x))/5+C