What is the integral of (tan(x))/(sec^3(x)) ?

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

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integral of (tan(x))/(sec^3(x))

\int \frac{tan ( x )}{sec ^{3} ( x )} d x

- \frac{cos ^{3} ( x )}{3} + C

Solution steps

\int \frac{tan ( x )}{sec ^{3} ( x )} d x

Express with sin, cos

= \int cos^{2} (x) sin (x) d x

Apply u-substitution

: \int - u^{2} d u

= \int - u^{2} d u

Take the constant out:

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

= - \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= - \frac{u ^{3}}{3}

Substitute back

u = cos (x)

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

Add a constant to the solution

= - \frac{cos ^{3} ( x )}{3} + C

\int \frac{x ^{4} + x ^{2} - 1}{x ^{3} + x} d x

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