What is the integral of (sin(2x))/(1+3cos^2(x)) ?

The integral of (sin(2x))/(1+3cos^2(x)) is -1/3 ln|1+3cos^2(x)|+C

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integral of (sin(2x))/(1+3cos^2(x))

\int \frac{sin ( 2 x )}{1 + 3 cos ^{2} ( x )} d x

- \frac{1}{3} ln 1 + 3 cos^{2} (x) + C

Solution steps

\int \frac{sin ( 2 x )}{1 + 3 cos ^{2} ( x )} d x

Apply u-substitution

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

Take the constant out:

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

= - \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{3} ln ∣ u ∣

Substitute back

u = 1 + 3 cos^{2} (x)

= - \frac{1}{3} ln 1 + 3 cos^{2} (x)

Add a constant to the solution

= - \frac{1}{3} ln 1 + 3 cos^{2} (x) + C

What is the integral of (sin(2x))/(1+3cos^2(x)) ?
The integral of (sin(2x))/(1+3cos^2(x)) is -1/3 ln|1+3cos^2(x)|+C