Solve the trigonometric integral $\int\mathrm{coth}\left(3x\right)dx$

Differentiate both sides of the equation $u=3x$

Find the derivative

The derivative of the linear function times a constant, is equal to the constant

The derivative of the linear function is equal to $1$

Rearrange the equation

Divide both sides of the equation by $3$

Taking the derivative of hyperbolic cotangent

The derivative of the linear function is equal to $1$

The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function

Any expression multiplied by $1$ is equal to itself

Replace $u$ with the value that we assigned to it in the beginning: $3x$

Replace $u$ with the value that we assigned to it in the beginning: $3x$

We can solve the integral $\int u\mathrm{csch}\left(u\right)^2du$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

First, identify or choose $u$ and calculate it's derivative, $du$

Now, identify $dv$ and calculate $v$

Solve the integral to find $v$

Apply the formula: $\int\mathrm{csch}\left(\theta \right)^2dx$$=-\mathrm{coth}\left(\theta \right)+C$, where $x=u$

Now replace the values of $u$, $du$ and $v$ in the last formula

Multiply the single term $\frac{1}{3}$ by each term of the polynomial $\left(-u\mathrm{coth}\left(u\right)+\int\mathrm{coth}\left(u\right)du\right)$

Replace $u$ with the value that we assigned to it in the beginning: $3x$

Multiply the fraction and term in $3-\frac{1}{3}x\mathrm{coth}\left(3x\right)$

Apply the trigonometric identity: $\mathrm{coth}\left(\theta \right)$$=\frac{\mathrm{cosh}\left(\theta \right)}{\mathrm{sinh}\left(\theta \right)}$, where $x=u$

We can solve the integral $\int\frac{\mathrm{cosh}\left(u\right)}{\mathrm{sinh}\left(u\right)}du$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $v$), which when substituted makes the integral easier. We see that $\mathrm{sinh}\left(u\right)$ it's a good candidate for substitution. Let's define a variable $v$ and assign it to the choosen part

Now, in order to rewrite $du$ in terms of $dv$, we need to find the derivative of $v$. We need to calculate $dv$, we can do that by deriving the equation above

Isolate $du$ in the previous equation

Substituting $v$ and $du$ in the integral and simplify

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

Replace $v$ with the value that we assigned to it in the beginning: $\mathrm{sinh}\left(u\right)$

Replace $u$ with the value that we assigned to it in the beginning: $3x$