What is the integral from 0 to pi of 6cos^2(3x) ?

The integral from 0 to pi of 6cos^2(3x) is 3pi

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integral from 0 to pi of 6cos^2(3x)

\int_{0}^{π} 6 cos^{2} (3 x) d x

3 π

Decimal Notation

9.42477 \dots

Solution steps

\int_{0}^{π} 6 cos^{2} (3 x) d x

Take the constant out:

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

= 6 \cdot \int_{0}^{π} cos^{2} (3 x) d x

Rewrite using trig identities

= 6 \cdot \int_{0}^{π} \frac{1}{2} (1 + cos (6 x)) d x

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int_{0}^{π} 1 + cos (6 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

= 6 \cdot \frac{1}{2} (\int_{0}^{π} 1 d x + \int_{0}^{π} cos (6 x) d x)

\int_{0}^{π} 1 d x = π

\int_{0}^{π} cos (6 x) d x = 0

= 6 \cdot \frac{1}{2} (π + 0)

Simplify

6 \cdot \frac{1}{2} (π + 0) : 3 π

= 3 π

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What is the integral from 0 to pi of 6cos^2(3x) ?
The integral from 0 to pi of 6cos^2(3x) is 3pi