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

The integral of (sec^2(x))/(tan^5(x)) is -1/4 cot^4(x)+C

integral of (sec^2(x))/(tan^5(x))

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

- \frac{1}{4} cot^{4} (x) + C

Solution steps

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

Rewrite using trig identities

= \int (1 + tan^{2} (x)) \frac{1}{tan ^{5} ( x )} d x

Apply u-substitution

= \int \frac{1}{u ^{5}} d u

Apply the Power Rule

= - \frac{1}{4 u ^{4}}

Substitute back

u = tan (x)

= - \frac{1}{4 tan ^{4} ( x )}

Simplify

- \frac{1}{4 tan ^{4} ( x )} : - \frac{1}{4} cot^{4} (x)

= - \frac{1}{4} cot^{4} (x)

Add a constant to the solution

= - \frac{1}{4} cot^{4} (x) + C

\int \frac{1}{sec ^{3} ( y )} d y

\int_{0}^{6} x^{2} d x

d er i v a t i v e sec (x) - 22 tan (x)

\frac{\partial}{\partial x} (4 y^{2} e^{7 x})

y = \frac{8 e ^{8 x}}{7 x + 2}

What is the integral of (sec^2(x))/(tan^5(x)) ?
The integral of (sec^2(x))/(tan^5(x)) is -1/4 cot^4(x)+C