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

The integral of ((csc^2(x)))/(cot^3(x)) is 1/2 tan^2(x)+C

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integral of ((csc^2(x)))/(cot^3(x))

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{u ^{3}} d u

= \int - \frac{1}{u ^{3}} d u

Take the constant out:

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

= - \int \frac{1}{u ^{3}} d u

Apply the Power Rule

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

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

Substitute back

u = cot (x)

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

Simplify

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

= \frac{1}{2} tan^{2} (x)

Add a constant to the solution

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

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What is the integral of ((csc^2(x)))/(cot^3(x)) ?
The integral of ((csc^2(x)))/(cot^3(x)) is 1/2 tan^2(x)+C