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

The integral from 0 to pi/2 of 3sin^2(3x) is (3pi)/4

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

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

\frac{3 π}{4}

Decimal

2.35619 \dots

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 3 \cdot \int_{0}^{\frac{π}{2}} \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

= 3 \cdot \frac{1}{2} \cdot \int_{0}^{\frac{π}{2}} 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

= 3 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d x - \int_{0}^{\frac{π}{2}} cos (6 x) d x)

\int_{0}^{\frac{π}{2}} 1 d x = \frac{π}{2}

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

= 3 \cdot \frac{1}{2} (\frac{π}{2} - 0)

Simplify

3 \cdot \frac{1}{2} (\frac{π}{2} - 0) : \frac{3 π}{4}

= \frac{3 π}{4}

\int \frac{3 x ^{2} + 5}{( x ^{2} + 4 ) ^{2}} d x

\int_{0}^{3} \frac{x}{36 - x ^{2}} d x

\int \frac{1 + x}{2 x} d x

y = x, y = 12 - 4 x, x = 0, x = 3

y^{^{'}} = \frac{4 x ^{3} + 1}{2 y - 6}, y (1) = 2

What is the integral from 0 to pi/2 of 3sin^2(3x) ?
The integral from 0 to pi/2 of 3sin^2(3x) is (3pi)/4