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

The integral of (cos(x))/(2+3sin^2(x)) is 1/(sqrt(6))arctan(sqrt(3/2)sin(x))+C

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

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

\frac{1}{6} arctan (\frac{3}{2} sin (x)) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 + 3 u ^{2}} d u

= \int \frac{1}{2 + 3 u ^{2}} d u

Apply Integral Substitution

= \int \frac{1}{6 ( 1 + v ^{2} )} d v

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

\int \frac{1}{1 + v ^{2}} d v = arctan (v)

= \frac{1}{6} arctan (v)

Substitute back

= \frac{1}{6} arctan (\frac{3}{2} sin (x))

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{1}{6} arctan (\frac{3}{2} sin (x))

Add a constant to the solution

= \frac{1}{6} arctan (\frac{3}{2} sin (x)) + C

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What is the integral of (cos(x))/(2+3sin^2(x)) ?
The integral of (cos(x))/(2+3sin^2(x)) is 1/(sqrt(6))arctan(sqrt(3/2)sin(x))+C