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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{csc ^{2} ( v )}{3} d v

Take the constant out:

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

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

Use the common integral:

\int csc^{2} (v) d v = - cot (v)

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

Substitute back

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

Simplify

2 \cdot \frac{1}{3} (- cot ((x)^{3})) : - \frac{2}{3} cot (x^{\frac{3}{2}})

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

Add a constant to the solution

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

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