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

The integral of (cos(u))/(sin^3(u)) is -1/2 csc^2(u)+C

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integral of (cos(u))/(sin^3(u))

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

- \frac{1}{2} csc^{2} (u) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

= \int - \frac{1}{2} d v

Integral of a constant:

\int a d x = a x

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

Substitute back

v = csc^{2} (u)

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

Simplify

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

Add a constant to the solution

= - \frac{1}{2} csc^{2} (u) + C

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\int sin (2 x) sec (2 x) d x

What is the integral of (cos(u))/(sin^3(u)) ?
The integral of (cos(u))/(sin^3(u)) is -1/2 csc^2(u)+C