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

The integral of (sin(2x))/((1+cos(2x))^2) is 1/4 sec^2(x)+C

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integral of (sin(2x))/((1+cos(2x))^2)

\int \frac{sin ( 2 x )}{( 1 + cos ( 2 x ) ) ^{2}} d x

\frac{1}{4} sec^{2} (x) + C

Solution steps

\int \frac{sin ( 2 x )}{( 1 + cos ( 2 x ) ) ^{2}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

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

Substitute back

u = 1 + cos (2 x)

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

Simplify

- \frac{1}{2} (- \frac{1}{1 + cos ( 2 x )}) : \frac{1}{4} sec^{2} (x)

= \frac{1}{4} sec^{2} (x)

Add a constant to the solution

= \frac{1}{4} sec^{2} (x) + C

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What is the integral of (sin(2x))/((1+cos(2x))^2) ?
The integral of (sin(2x))/((1+cos(2x))^2) is 1/4 sec^2(x)+C