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

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

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

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

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

Solution steps

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

Rewrite using trig identities

= \int \frac{x ^{2} cos ( x ^{3} - 2 )}{sin ^{2} ( x ^{3} - 2 )} d x

Apply u-substitution

: \int \frac{cot ( u )}{3 sin ( u )} d u

= \int \frac{cot ( u )}{3 sin ( u )} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{cot ( u )}{sin ( u )} d u

Rewrite using trig identities

= \frac{1}{3} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

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

= \frac{1}{3} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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