What is the integral of (sin(x))/((cos(x))^3) ?

The integral of (sin(x))/((cos(x))^3) is 1/2 sec^2(x)+C

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integral of (sin(x))/((cos(x))^3)

\int \frac{sin ( x )}{( cos ( x ) ) ^{3}} d x

\frac{1}{2} sec^{2} (x) + C

Solution steps

\int \frac{sin ( x )}{( cos ( x ) ) ^{3}} d x

Apply u-substitution

: \int - \frac{1}{u ^{3}} d u

= \int - \frac{1}{u ^{3}} d u

Take the constant out:

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

= - \int \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= - (- \frac{1}{2 u ^{2}})

Substitute back

u = cos (x)

= - (- \frac{1}{2 cos ^{2} ( x )})

Simplify

- (- \frac{1}{2 cos ^{2} ( x )}) : \frac{1}{2} sec^{2} (x)

= \frac{1}{2} sec^{2} (x)

Add a constant to the solution

= \frac{1}{2} sec^{2} (x) + C

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What is the integral of (sin(x))/((cos(x))^3) ?
The integral of (sin(x))/((cos(x))^3) is 1/2 sec^2(x)+C