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

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

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integral of coth^2(3x)

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

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

Solution steps

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

Rewrite using trig identities

= \int 1 + csch^{2} (3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x + \int csch^{2} (3 x) d x

\int 1 d x = x

\int csch^{2} (3 x) d x = - \frac{1}{3} coth (3 x)

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

Add a constant to the solution

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

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