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

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

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integral of (sin(x))/(sqrt(1+sin^2(x)))

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

- arcsin (\frac{1}{2} cos (x)) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 1 d v

= - \int 1 d v

Integral of a constant:

\int a d x = a x

= - 1 \cdot v

Substitute back

= - 1 \cdot arcsin (\frac{1}{2} cos (x))

Simplify

= - arcsin (\frac{1}{2} cos (x))

Add a constant to the solution

= - arcsin (\frac{1}{2} cos (x)) + C

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