integral from 0 to 2 of xe^{(-x^2-1)/2}

Lösung

\int_{0}^{2} x e^{\frac{- x ^{2} - 1}{2}} d x

\frac{1}{e} - \frac{1}{e ^{\frac{5}{2}}}

Dezimale

0.52444 \dots

Schritte zur Lösung

\int_{0}^{2} x e^{\frac{- x ^{2} - 1}{2}} d x

Wende U-Substitution an

= \int_{- 1}^{- 5} - \frac{e ^{\frac{u}{2}}}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 5}^{- 1} - \frac{e ^{\frac{u}{2}}}{2} d u

Entferne die Konstante:

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

= - (- \frac{1}{2} \cdot \int_{- 5}^{- 1} e^{\frac{u}{2}} d u)

Wende U-Substitution an

= - (- \frac{1}{2} \cdot \int_{- \frac{5}{2}}^{- \frac{1}{2}} e^{v} \cdot 2 d v)

Entferne die Konstante:

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

= - (- \frac{1}{2} \cdot 2 \cdot \int_{- \frac{5}{2}}^{- \frac{1}{2}} e^{v} d v)

Nutze das gemeinsame Integral :

\int e^{v} d v = e^{v}

= - (- \frac{1}{2} \cdot 2 [e^{v}]_{- \frac{5}{2}}^{- \frac{1}{2}})

Vereinfache

- (- \frac{1}{2} \cdot 2 [e^{v}]_{- \frac{5}{2}}^{- \frac{1}{2}}) : [e^{v}]_{- \frac{5}{2}}^{- \frac{1}{2}}

= [e^{v}]_{- \frac{5}{2}}^{- \frac{1}{2}}

Berechne die Grenzen:

\frac{1}{e} - \frac{1}{e ^{\frac{5}{2}}}

= \frac{1}{e} - \frac{1}{e ^{\frac{5}{2}}}

\int_{1}^{4} - x - (2 - x) d x

\int_{1}^{4} (2 x) d x

\int_{0}^{π} cos (\frac{1}{2}) x d x

x \to \frac{π}{2} lim (\frac{cos ( x )}{1 - sin ( x )})

x \to \frac{π}{2} lim (\frac{ln ( sin ( x ) )}{( π - 2 x ) ^{2}})