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

The integral from 0 to 2pi of x^2cos(x) is 4pi

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

\int_{0}^{2 π} x^{2} cos (x) d x

4 π

Decimal

12.56637 \dots

Solution steps

\int_{0}^{2 π} x^{2} cos (x) d x

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

4 π

= 4 π

\int_{0}^{1} x^{2} (4 x^{3} - 1)^{4} d x

\int_{0}^{1} y (y^{2} + 1)^{5} d y

\int_{1}^{3} (x + 2) d x

\int_{0}^{1} \frac{2}{( 2 x - 1 ) ^{2}} d x

\int_{- 1}^{1} \frac{1}{4 x ^{2} - 9} d x

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