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

The integral of 2tan^2(2x) is -2x+tan(2x)+C

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integral of 2tan^2(2x)

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

- 2 x + tan (2 x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int tan^{2} (2 x) d x

Rewrite using trig identities

= 2 \cdot \int - 1 + sec^{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

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

\int 1 d x = x

\int sec^{2} (2 x) d x = \frac{1}{2} tan (2 x)

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

Simplify

2 (- x + \frac{1}{2} tan (2 x)) : - 2 x + tan (2 x)

= - 2 x + tan (2 x)

Add a constant to the solution

= - 2 x + tan (2 x) + C

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What is the integral of 2tan^2(2x) ?
The integral of 2tan^2(2x) is -2x+tan(2x)+C