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Beliebt Rechnen >

integral of (x^2-1)/(e^{x/2)}

Lösung

\int \frac{x ^{2} - 1}{e ^{\frac{x}{2}}} d x

Lösung

2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}}) + C

Schritte zur Lösung

\int \frac{x ^{2} - 1}{e ^{\frac{x}{2}}} d x

Wende U-Substitution an

= \int \frac{2 ( 4 u ^{2} - 1 )}{e ^{u}} d u

Entferne die Konstante:

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

= 2 \cdot \int \frac{4 u ^{2} - 1}{e ^{u}} d u

Multipliziere aus

\frac{4 u ^{2} - 1}{e ^{u}} : \frac{4 u ^{2}}{e ^{u}} - \frac{1}{e ^{u}}

= 2 \cdot \int \frac{4 u ^{2}}{e ^{u}} - \frac{1}{e ^{u}} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{4 u ^{2}}{e ^{u}} d u - \int \frac{1}{e ^{u}} d u)

\int \frac{4 u ^{2}}{e ^{u}} d u = 4 (- \frac{u ^{2}}{e ^{u}} + 2 (- \frac{u}{e ^{u}} - \frac{1}{e ^{u}}))

\int \frac{1}{e ^{u}} d u = - \frac{1}{e ^{u}}

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

Setze in

u = \frac{x}{2}

ein

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

Vereinfache

2 (4 (- \frac{( \frac{x}{2} ) ^{2}}{e ^{\frac{x}{2}}} + 2 (- \frac{\frac{x}{2}}{e ^{\frac{x}{2}}} - \frac{1}{e ^{\frac{x}{2}}})) - (- \frac{1}{e ^{\frac{x}{2}}})) : 2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}})

= 2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}})

Füge eine Konstante zur Lösung hinzu

= 2 (- \frac{x ^{2}}{e ^{\frac{x}{2}}} - \frac{4 x}{e ^{\frac{x}{2}}} - \frac{7}{e ^{\frac{x}{2}}}) + C

Graph

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