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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{u ^{2}} d u

= 2 \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

u = 1 + tan (x)

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

Simplify

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

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

Add a constant to the solution

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

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