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

The integral of-(sin(x))/(cos^3(x)) is -(tan^2(x))/2+C

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

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

- \frac{tan ^{2} ( x )}{2} + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

= - \int u d u

Apply the Power Rule

= - \frac{u ^{2}}{2}

Substitute back

u = tan (x)

= - \frac{tan ^{2} ( x )}{2}

Add a constant to the solution

= - \frac{tan ^{2} ( x )}{2} + C

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