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

The integral of y^2csc^2(y^3)cot^2(y^3) is -1/9 cot^3(y^3)+C

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

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

- \frac{1}{9} cot^{3} (y^{3}) + C

Solution steps

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

Apply u-substitution

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

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

Apply u-substitution

: \int - v^{2} d v

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{v ^{3}}{3}

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

Substitute back

= \frac{1}{3} (- \frac{cot ^{3} ( y ^{3} )}{3})

Simplify

\frac{1}{3} (- \frac{cot ^{3} ( y ^{3} )}{3}) : - \frac{1}{9} cot^{3} (y^{3})

= - \frac{1}{9} cot^{3} (y^{3})

Add a constant to the solution

= - \frac{1}{9} cot^{3} (y^{3}) + C

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What is the integral of y^2csc^2(y^3)cot^2(y^3) ?
The integral of y^2csc^2(y^3)cot^2(y^3) is -1/9 cot^3(y^3)+C