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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{u ^{2} + 4} d u

= \int \frac{1}{u ^{2} + 4} d u

Apply Integral Substitution

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

Substitute back

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

Add a constant to the solution

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

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