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

The integral of 2cos^2(2x) is x+1/4 sin(4x)+C

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integral of 2cos^2(2x)

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

x + \frac{1}{4} sin (4 x) + C

Solution steps

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

Rewrite using trig identities

= 2 \cdot \int \frac{1}{2} (1 + cos (4 x)) d x

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 1 + cos (4 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 \cdot \frac{1}{2} (\int 1 d x + \int cos (4 x) d x)

\int 1 d x = x

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

= 2 \cdot \frac{1}{2} (x + \frac{1}{4} sin (4 x))

Simplify

2 \cdot \frac{1}{2} (x + \frac{1}{4} sin (4 x)) : x + \frac{1}{4} sin (4 x)

= x + \frac{1}{4} sin (4 x)

Add a constant to the solution

= x + \frac{1}{4} sin (4 x) + C

What is the integral of 2cos^2(2x) ?
The integral of 2cos^2(2x) is x+1/4 sin(4x)+C