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

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

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

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

- \frac{1}{3} arctan (\frac{cos ( x )}{3}) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integral Substitution

= - \int \frac{1}{3 ( v ^{2} + 1 )} d v

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}{v ^{2} + 1} d v

Use the common integral:

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

= - \frac{1}{3} arctan (v)

Substitute back

= - \frac{1}{3} arctan (\frac{cos ( x )}{3})

Add a constant to the solution

= - \frac{1}{3} arctan (\frac{cos ( x )}{3}) + C

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