Solve \(\int_{0}^{2 \pi} x^{2} \cos x dx\)

Question

Evaluate the integral

4 π

Alternative Form

\approx 12.566371

Evaluate

\int_{0}^{2 π} x^{2} cos (x) d x

Evaluate the integral

\int x^{2} cos (x) d x

Prepare for integration by parts

u = x^{2} d v = cos (x) d x

Calculate the derivative

More Steps

d u = 2 x d x d v = cos (x) d x

Evaluate the integral

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d u = 2 x d x v = sin (x)

Substitute u = x^{2} 、 v = sin (x) 、 d u = 2 x d x 、 d v = cos (x) d x for \int u d v = uv - \int v d u

x^{2} sin (x) - \int 2 x sin (x) d x

Evaluate the integral

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

Return the limits

(x^{2} sin (x) + 2 x cos (x) - 2 sin (x))_{0}^{2 π}

Solution

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4 π

Alternative Form

\approx 12.566371

Show Solution