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

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

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

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

- csc^{2} (x) + C

Solution steps

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

Rewrite using trig identities

= 2 \cdot \int cot (x) csc^{2} (x) d x

Apply u-substitution

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

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

Integral of a constant:

\int a d x = a x

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

Substitute back

u = csc^{2} (x)

= 2 (- \frac{1}{2}) csc^{2} (x)

Simplify

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

= - csc^{2} (x)

Add a constant to the solution

= - csc^{2} (x) + C

x cos (x - 3)

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\frac{d}{d x} (3 x^{2} + 4 x - 5)

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