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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Integral of a constant:

\int a d x = a x

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

Substitute back

u = csc^{2} (x)

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

Simplify

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

Add a constant to the solution

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

\int \frac{x - 6}{x ^{2} - 18 x + 82} d x

\int t^{2} e^{- 17 t} d t

x \to 3 lim (9 x + 92)

\frac{d}{d x} ((f (x))^{n})

\int e^{- 18 x} d x

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