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

The integral from 0 to pi of 3x^2cos(x/2) is 3(2pi^2-16)

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

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

3 (2 π^{2} - 16)

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

: [2 x^{2} sin (\frac{x}{2}) - \int 4 x sin (\frac{x}{2}) d x]_{0}^{π}

= 3 [2 x^{2} sin (\frac{x}{2}) - \int 4 x sin (\frac{x}{2}) d x]_{0}^{π}

\int 4 x sin (\frac{x}{2}) d x = 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}))

= 3 [2 x^{2} sin (\frac{x}{2}) - 4 (- 2 x cos (\frac{x}{2}) + 4 sin (\frac{x}{2}))]_{0}^{π}

Compute the boundaries:

2 π^{2} - 16

= 3 (2 π^{2} - 16)

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What is the integral from 0 to pi of 3x^2cos(x/2) ?
The integral from 0 to pi of 3x^2cos(x/2) is 3(2pi^2-16)