What is the integral of tanh^2(2x) ?

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

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integral of tanh^2(2x)

\int tanh^{2} (2 x) d x

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

Solution steps

\int tanh^{2} (2 x) d x

Apply u-substitution

: \int tanh^{2} (u) \frac{1}{2} d u

= \int tanh^{2} (u) \frac{1}{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 tanh^{2} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int 1 - sech^{2} (u) d u

Apply the Sum Rule:

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

= \frac{1}{2} (\int 1 d u - \int sech^{2} (u) d u)

\int 1 d u = u

\int sech^{2} (u) d u = tanh (u)

= \frac{1}{2} (u - tanh (u))

Substitute back

u = 2 x

= \frac{1}{2} (2 x - tanh (2 x))

Simplify

\frac{1}{2} (2 x - tanh (2 x)) : x - \frac{1}{2} tanh (2 x)

= x - \frac{1}{2} tanh (2 x)

Add a constant to the solution

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

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