What is the integral from 0 to 2 of x^3sin(x) ?

The integral from 0 to 2 of x^3sin(x) is 4cos(2)+6sin(2)

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integral from 0 to 2 of x^3sin(x)

\int_{0}^{2} x^{3} sin (x) d x

4 cos (2) + 6 sin (2)

Decimal

3.79119 \dots

Solution steps

\int_{0}^{2} x^{3} sin (x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

4 cos (2) + 6 sin (2)

= 4 cos (2) + 6 sin (2)

\int_{0}^{3} e^{x^{2}} x d x

\int_{0}^{2} 8 x - x^{2} d x

\int_{- 1}^{2} 3 x + 6 d x

\int_{- π}^{π} (1 + x) sin (n x) d x

\int_{0}^{5} (2 e^{x}) d x

What is the integral from 0 to 2 of x^3sin(x) ?
The integral from 0 to 2 of x^3sin(x) is 4cos(2)+6sin(2)