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

The integral of x^2csch^2(x^3) is -1/3 coth(x^3)+C

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integral of x^2csch^2(x^3)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{u csch ^{2} ( u ^{\frac{3}{2}} )}{2 x} d u

= \int \frac{csch ^{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 csch^{2} (u) d u

Use the common integral:

\int csch^{2} (u) d u = - coth (u)

= \frac{1}{3} (- coth (u))

Substitute back

u = x^{3}

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

Simplify

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

Add a constant to the solution

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

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