What is the integral of xcot^3(x^2-1)csc(x^2-1) ?

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

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integral of xcot^3(x^2-1)csc(x^2-1)

\int x cot^{3} (x^{2} - 1) csc (x^{2} - 1) d x

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

Solution steps

\int x cot^{3} (x^{2} - 1) csc (x^{2} - 1) d x

Apply u-substitution

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

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

Take the constant out:

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

= \frac{1}{2} \cdot \int cot^{3} (u) csc (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

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

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

Apply the Sum Rule:

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

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

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

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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