What is the integral from 0 to pi/2 of 4xe^xsin(x) ?

The integral from 0 to pi/2 of 4xe^xsin(x) is pie^{pi/2}-2

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integral from 0 to pi/2 of 4xe^xsin(x)

\int_{0}^{\frac{π}{2}} 4 x e^{x} sin (x) d x

π e^{\frac{π}{2}} - 2

Decimal

13.11256 \dots

Solution steps

\int_{0}^{\frac{π}{2}} 4 x e^{x} sin (x) d x

Take the constant out:

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

= 4 \cdot \int_{0}^{\frac{π}{2}} x e^{x} sin (x) d x

Apply Integration By Parts

: [x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - \int \frac{1}{2} (- e^{x} cos (x) + e^{x} sin (x)) d x]_{0}^{\frac{π}{2}}

= 4 [x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - \int \frac{1}{2} (- e^{x} cos (x) + e^{x} sin (x)) d x]_{0}^{\frac{π}{2}}

\int \frac{1}{2} (- e^{x} cos (x) + e^{x} sin (x)) d x = - \frac{1}{2} e^{x} cos (x)

= 4 [x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - (- \frac{1}{2} e^{x} cos (x))]_{0}^{\frac{π}{2}}

Simplify

4 [x (- \frac{1}{2} e^{x} cos (x) + \frac{1}{2} e^{x} sin (x)) - (- \frac{1}{2} e^{x} cos (x))]_{0}^{\frac{π}{2}} : 4 [\frac{1}{2} (- e^{x} x cos (x) + e^{x} x sin (x) + e^{x} cos (x))]_{0}^{\frac{π}{2}}

= 4 [\frac{1}{2} (- e^{x} x cos (x) + e^{x} x sin (x) + e^{x} cos (x))]_{0}^{\frac{π}{2}}

Compute the boundaries:

\frac{π e ^{\frac{π}{2}}}{4} - \frac{1}{2}

= 4 (\frac{π e ^{\frac{π}{2}}}{4} - \frac{1}{2})

Simplify

= π e^{\frac{π}{2}} - 2

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What is the integral from 0 to pi/2 of 4xe^xsin(x) ?
The integral from 0 to pi/2 of 4xe^xsin(x) is pie^{pi/2}-2