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

Lösung

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

\frac{3 e ^{\frac{π}{2}} - 2}{5}

Dezimale

2.48628 \dots

Schritte zur Lösung

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

Wende die partielle Integration an

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

\int sin (2 x) e^{x} d x = - \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5}

= [sin^{2} (x) e^{x} - (- \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5})]_{0}^{\frac{π}{2}}

Vereinfache

[sin^{2} (x) e^{x} - (- \frac{2 e ^{x} cos ( 2 x )}{5} + \frac{e ^{x} sin ( 2 x )}{5})]_{0}^{\frac{π}{2}} : [\frac{1}{5} (5 e^{x} sin^{2} (x) + 2 e^{x} cos (2 x) - e^{x} sin (2 x))]_{0}^{\frac{π}{2}}

= [\frac{1}{5} (5 e^{x} sin^{2} (x) + 2 e^{x} cos (2 x) - e^{x} sin (2 x))]_{0}^{\frac{π}{2}}

Berechne die Grenzen:

\frac{3 e ^{\frac{π}{2}}}{5} - \frac{2}{5}

= \frac{3 e ^{\frac{π}{2}}}{5} - \frac{2}{5}

Vereinfache

= \frac{3 e ^{\frac{π}{2}} - 2}{5}

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