What is the integral from 1 to 2 of xsin(sqrt(x)) ?

The integral from 1 to 2 of xsin(sqrt(x)) is 2(4sqrt(2)cos(sqrt(2))-5cos(1)+3sin(1))

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integral from 1 to 2 of xsin(sqrt(x))

\int_{1}^{2} x sin (x) d x

2 (42 cos (2) - 5 cos (1) + 3 sin (1))

Decimal Notation

1.41010 \dots

Solution steps

\int_{1}^{2} x sin (x) d x

Apply u-substitution

: \int_{1}^{2} 2 u^{3} sin (u) d u

= \int_{1}^{2} 2 u^{3} sin (u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{1}^{2} u^{3} sin (u) d u

Apply Integration By Parts

: [- u^{3} cos (u) - \int - 3 u^{2} cos (u) d u]_{1}^{2}

= 2 [- u^{3} cos (u) - \int - 3 u^{2} cos (u) d u]_{1}^{2}

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

= 2 [- u^{3} cos (u) - (- 3 (u^{2} sin (u) - 2 (- u cos (u) + sin (u))))]_{1}^{2}

Simplify

= 2 [- u^{3} cos (u) + 3 (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))]_{1}^{2}

Compute the boundaries:

42 cos (2) - 5 cos (1) + 3 sin (1)

= 2 (42 cos (2) - 5 cos (1) + 3 sin (1))

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What is the integral from 1 to 2 of xsin(sqrt(x)) ?
The integral from 1 to 2 of xsin(sqrt(x)) is 2(4sqrt(2)cos(sqrt(2))-5cos(1)+3sin(1))