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

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

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integral of 2sec^2(2x)

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

tan (2 x) + C

Solution steps

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

Apply u-substitution

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

= 2 \cdot \int sec^{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

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

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= 2 \cdot \frac{1}{2} tan (u)

Substitute back

u = 2 x

= 2 \cdot \frac{1}{2} tan (2 x)

Simplify

2 \cdot \frac{1}{2} tan (2 x) : tan (2 x)

= tan (2 x)

Add a constant to the solution

= tan (2 x) + C

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