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

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

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

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

2 π^{2} - 16

Decimal

3.73920 \dots

Solution steps

\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}

= [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}))

= [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

= 2 π^{2} - 16

\int_{- 2}^{2} (∣ x ∣) - (x^{2} - 2) d x

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

\int_{0}^{t} u e^{2 u} cos (3 u) d u

\int_{0}^{π} x ∣ cos (x) ∣ d x

\int_{0}^{1} x \frac{x}{1 - x} d x

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