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

The integral of (sin(2x))/(sin^3(x)) is -2csc(x)+C

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

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

- 2 csc (x) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= 2 (- \frac{1}{u})

Substitute back

u = sin (x)

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

Simplify

2 (- \frac{1}{sin ( x )}) : - 2 csc (x)

= - 2 csc (x)

Add a constant to the solution

= - 2 csc (x) + C

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What is the integral of (sin(2x))/(sin^3(x)) ?
The integral of (sin(2x))/(sin^3(x)) is -2csc(x)+C