What is the integral of (-sec^2(x))/(tan^7(x)) ?

The integral of (-sec^2(x))/(tan^7(x)) is 1/6 cot^6(x)+C

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integral of (-sec^2(x))/(tan^7(x))

\int \frac{- sec ^{2} ( x )}{tan ^{7} ( x )} d x

\frac{1}{6} cot^{6} (x) + C

Solution steps

\int \frac{- sec ^{2} ( x )}{tan ^{7} ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{sec ^{2} ( x )}{tan ^{7} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{7}} d u

= - \int \frac{1}{u ^{7}} d u

Apply the Power Rule

: - \frac{1}{6 u ^{6}}

= - (- \frac{1}{6 u ^{6}})

Substitute back

u = tan (x)

= - (- \frac{1}{6 tan ^{6} ( x )})

Simplify

- (- \frac{1}{6 tan ^{6} ( x )}) : \frac{1}{6} cot^{6} (x)

= \frac{1}{6} cot^{6} (x)

Add a constant to the solution

= \frac{1}{6} cot^{6} (x) + C

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What is the integral of (-sec^2(x))/(tan^7(x)) ?
The integral of (-sec^2(x))/(tan^7(x)) is 1/6 cot^6(x)+C