What is the integral of xcsch(x^2)coth(x^2) ?

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

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integral of xcsch(x^2)coth(x^2)

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

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

Solution steps

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

Apply u-substitution

: \int \frac{csch ( u ) coth ( u )}{2} d u

= \int \frac{csch ( u ) coth ( 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 csch (u) coth (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int \frac{cosh ( u )}{sinh ^{2} ( u )} d u

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{v}

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

Substitute back

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

Simplify

\frac{1}{2} (- \frac{1}{sinh ( x ^{2} )}) : - \frac{1}{2} csch (x^{2})

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

Add a constant to the solution

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

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