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

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

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

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

2 (- \frac{1}{3} csc^{3} (x) + csc (x)) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 2 \cdot \int \frac{( 1 - sin ^{2} ( x ) ) cos ( x )}{sin ^{4} ( x )} d x

Apply u-substitution

: \int \frac{1 - u ^{2}}{u ^{4}} d u

= 2 \cdot \int \frac{1 - u ^{2}}{u ^{4}} d u

Expand

\frac{1 - u ^{2}}{u ^{4}} : \frac{1}{u ^{4}} - \frac{1}{u ^{2}}

= 2 \cdot \int \frac{1}{u ^{4}} - \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

= 2 (\int \frac{1}{u ^{4}} d u - \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

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

= 2 (- \frac{1}{3 u ^{3}} - (- \frac{1}{u}))

Substitute back

u = sin (x)

= 2 (- \frac{1}{3 sin ^{3} ( x )} - (- \frac{1}{sin ( x )}))

Simplify

2 (- \frac{1}{3 sin ^{3} ( x )} - (- \frac{1}{sin ( x )})) : 2 (- \frac{1}{3} csc^{3} (x) + csc (x))

= 2 (- \frac{1}{3} csc^{3} (x) + csc (x))

Add a constant to the solution

= 2 (- \frac{1}{3} csc^{3} (x) + csc (x)) + C

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