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

The integral of (sin(2x))/(sqrt(1+cos^2(x))) is -2sqrt(1+cos^2(x))+C

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

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

- 2 1 + cos^{2} (x) + C

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = 1 + cos^{2} (x)

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

Apply radical rule:

a^{\frac{1}{n}} = n a

= - 2 1 + cos^{2} (x)

Add a constant to the solution

= - 2 1 + cos^{2} (x) + C

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