What is the integral from 0 to 3pi of 2cos^2(x/3) ?

The integral from 0 to 3pi of 2cos^2(x/3) is 3pi

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integral from 0 to 3pi of 2cos^2(x/3)

\int_{0}^{3 π} 2 cos^{2} (\frac{x}{3}) d x

3 π

Decimal

9.42477 \dots

Solution steps

\int_{0}^{3 π} 2 cos^{2} (\frac{x}{3}) d x

Take the constant out:

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

= 2 \cdot \int_{0}^{3 π} cos^{2} (\frac{x}{3}) d x

Rewrite using trig identities

= 2 \cdot \int_{0}^{3 π} \frac{1}{2} (1 + cos (\frac{2 x}{3})) 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_{0}^{3 π} 1 + cos (\frac{2 x}{3}) 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_{0}^{3 π} 1 d x + \int_{0}^{3 π} cos (\frac{2 x}{3}) d x)

\int_{0}^{3 π} 1 d x = 3 π

\int_{0}^{3 π} cos (\frac{2 x}{3}) d x = 0

= 2 \cdot \frac{1}{2} (3 π + 0)

Simplify

2 \cdot \frac{1}{2} (3 π + 0) : 3 π

= 3 π

What is the integral from 0 to 3pi of 2cos^2(x/3) ?
The integral from 0 to 3pi of 2cos^2(x/3) is 3pi