What is the integral of (cot^3(x))/(csc(x)) ?

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

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integral of (cot^3(x))/(csc(x))

\int \frac{cot ^{3} ( x )}{csc ( x )} d x

- (csc (x) + sin (x)) + C

Solution steps

\int \frac{cot ^{3} ( x )}{csc ( x )} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Expand

\frac{u ^{2} - 1}{u ^{2}} : 1 - \frac{1}{u ^{2}}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

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

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

Substitute back

u = csc (x)

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

Simplify

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

= - (csc (x) + sin (x))

Add a constant to the solution

= - (csc (x) + sin (x)) + C

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