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

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

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integral of sech^2(2x)

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

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

Solution steps

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

Rewrite using trig identities

= \int 1 - tanh^{2} (2 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 tanh^{2} (2 x) d x

\int 1 d x = x

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

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

Simplify

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

Add a constant to the solution

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

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