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

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

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

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

- 2 π - 6

Decimal Notation

- 12.28318 \dots

Solution steps

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

Apply Integration By Parts

: [(x^{2} + 3 x) sin (x) - \int (2 x + 3) sin (x) d x]_{0}^{π}

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

\int (2 x + 3) sin (x) d x = - cos (x) (2 x + 3) + 2 sin (x)

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

Expand

(x^{2} + 3 x) sin (x) - (- cos (x) (2 x + 3) + 2 sin (x)) : x^{2} sin (x) + 3 x sin (x) + 2 x cos (x) + 3 cos (x) - 2 sin (x)

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

Compute the boundaries:

- 2 π - 6

= - 2 π - 6

\int_{s}^{s} 1 d x

\int_{0^{+}}^{1} \frac{1}{x ^{2}} d x

\int_{\infty}^{0} e^{2 x} d x

\int_{4}^{9} x x^{2} - 4 d x

\int_{0}^{l n (3)} e^{2 x} - e^{x} d x

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