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

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

\int csc (x) sec (x) d x

f (x) = 6 x

f (x) = 3 e^{(2 x^{2})}

(y - 9)^{2} = y^{^{'}}

\frac{t}{t ^{2} + 1}

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