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

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

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integral of (4sin(x))/((1+cos(x))^2)

\int \frac{4 sin ( x )}{( 1 + cos ( x ) ) ^{2}} d x

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

Solution steps

\int \frac{4 sin ( x )}{( 1 + cos ( x ) ) ^{2}} d x

Take the constant out:

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

= 4 \cdot \int \frac{sin ( x )}{( 1 + cos ( x ) ) ^{2}} d x

Apply u-substitution

= 4 \cdot \int u d u

Apply the Power Rule

= 4 \cdot \frac{u ^{2}}{2}

Substitute back

u = tan (\frac{x}{2})

= 4 \cdot \frac{tan ^{2} ( \frac{x}{2} )}{2}

Simplify

4 \cdot \frac{tan ^{2} ( \frac{x}{2} )}{2} : 2 tan^{2} (\frac{x}{2})

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

Add a constant to the solution

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

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