What is the integral of (sec^2(x))/((1+tan(x))^3) ?

The integral of (sec^2(x))/((1+tan(x))^3) is -1/(2(1+tan(x))^2)+C

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integral of (sec^2(x))/((1+tan(x))^3)

\int \frac{sec ^{2} ( x )}{( 1 + tan ( x ) ) ^{3}} d x

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

Solution steps

\int \frac{sec ^{2} ( x )}{( 1 + tan ( x ) ) ^{3}} d x

Apply u-substitution

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

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

Apply the Power Rule

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

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

Substitute back

u = 1 + tan (x)

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

Add a constant to the solution

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

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