What is the integral of (sin(x))/((cos(x))^5) ?

The integral of (sin(x))/((cos(x))^5) is 1/4 sec^4(x)+C

integral of (sin(x))/((cos(x))^5)

\int \frac{sin ( x )}{( cos ( x ) ) ^{5}} d x

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

Solution steps

\int \frac{sin ( x )}{( cos ( x ) ) ^{5}} d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = cos (x)

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

Simplify

- (- \frac{1}{4 cos ^{4} ( x )}) : \frac{1}{4} sec^{4} (x)

= \frac{1}{4} sec^{4} (x)

Add a constant to the solution

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

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What is the integral of (sin(x))/((cos(x))^5) ?
The integral of (sin(x))/((cos(x))^5) is 1/4 sec^4(x)+C