What is the integral from-pi/4 to pi/4 of cos^2(2x) ?

The integral from-pi/4 to pi/4 of cos^2(2x) is pi/4

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integral from-pi/4 to pi/4 of cos^2(2x)

\int_{- \frac{π}{4}}^{\frac{π}{4}} cos^{2} (2 x) d x

\frac{π}{4}

Decimal

0.78539 \dots

Solution steps

\int_{- \frac{π}{4}}^{\frac{π}{4}} cos^{2} (2 x) d x

Rewrite using trig identities

= \int_{- \frac{π}{4}}^{\frac{π}{4}} \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

= \frac{1}{2} \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} 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

= \frac{1}{2} (\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d x + \int_{- \frac{π}{4}}^{\frac{π}{4}} cos (4 x) d x)

\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d x = \frac{π}{2}

\int_{- \frac{π}{4}}^{\frac{π}{4}} cos (4 x) d x = 0

= \frac{1}{2} (\frac{π}{2} + 0)

Simplify

\frac{1}{2} (\frac{π}{2} + 0) : \frac{π}{4}

= \frac{π}{4}

What is the integral from-pi/4 to pi/4 of cos^2(2x) ?
The integral from-pi/4 to pi/4 of cos^2(2x) is pi/4