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

The integral from-2pi to 4pi of cos^2(x) is 3pi

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

\int_{- 2 π}^{4 π} cos^{2} (x) d x

3 π

Decimal

9.42477 \dots

Solution steps

\int_{- 2 π}^{4 π} cos^{2} (x) d x

Rewrite using trig identities

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

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

\int_{- 2 π}^{4 π} 1 d x = 6 π

\int_{- 2 π}^{4 π} cos (2 x) d x = 0

= \frac{1}{2} (6 π + 0)

Simplify

\frac{1}{2} (6 π + 0) : 3 π

= 3 π

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What is the integral from-2pi to 4pi of cos^2(x) ?
The integral from-2pi to 4pi of cos^2(x) is 3pi