What is the integral of (sin(x))/(2-sin^2(x)) ?

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

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integral of (sin(x))/(2-sin^2(x))

\int \frac{sin ( x )}{2 - sin ^{2} ( x )} d x

- arctan (cos (x)) + C

Solution steps

\int \frac{sin ( x )}{2 - sin ^{2} ( x )} d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Use the common integral:

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

= - arctan (u)

Substitute back

u = cos (x)

= - arctan (cos (x))

Add a constant to the solution

= - arctan (cos (x)) + C

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