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Beliebt Rechnen >

integral of (1/8)y_{1}e^{-(y_{1}+y_{2})/2}_{1}

Lösung

\int (\frac{1}{8}) y_{1} e^{- \frac{y _{1} + y _{2}}{2}} d y_{1}

Lösung

- \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2) + C

Schritte zur Lösung

\int \frac{1}{8} y_{1} e^{- \frac{y _{1} + y _{2}}{2}} d y_{1}

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int e^{- \frac{y _{2} + y _{1}}{2}} y_{1} d y_{1}

Multipliziere aus

- \frac{y _{2} + y _{1}}{2} : - \frac{y _{2}}{2} - \frac{y _{1}}{2}

= \frac{1}{8} \cdot \int e^{- \frac{y _{2}}{2} - \frac{y _{1}}{2}} y_{1} d y_{1}

Vereinfache

e^{- \frac{y _{2}}{2} - \frac{y _{1}}{2}} : e^{- \frac{y _{2}}{2}} e^{- \frac{y _{1}}{2}}

= \frac{1}{8} \cdot \int e^{- \frac{y _{2}}{2}} e^{- \frac{y _{1}}{2}} y_{1} d y_{1}

Entferne die Konstante:

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

= \frac{1}{8} e^{- \frac{y _{2}}{2}} \cdot \int e^{- \frac{y _{1}}{2}} y_{1} d y_{1}

Wende die partielle Integration an

= \frac{1}{8} e^{- \frac{y _{2}}{2}} (- 2 e^{- \frac{y _{1}}{2}} y_{1} - \int - 2 e^{- \frac{y _{1}}{2}} d y_{1})

\int - 2 e^{- \frac{y _{1}}{2}} d y_{1} = 4 e^{- \frac{y _{1}}{2}}

= \frac{1}{8} e^{- \frac{y _{2}}{2}} (- 2 e^{- \frac{y _{1}}{2}} y_{1} - 4 e^{- \frac{y _{1}}{2}})

Vereinfache

\frac{1}{8} e^{- \frac{y _{2}}{2}} (- 2 e^{- \frac{y _{1}}{2}} y_{1} - 4 e^{- \frac{y _{1}}{2}}) : - \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2)

= - \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{4} e^{\frac{- y _{1} - y _{2}}{2}} (y_{1} + 2) + C

Graph

Beliebte Beispiele

y^{^{'}} = 4 x y

integral from-2 to 1 of sqrt(1+25)

\int_{- 2}^{1} 1 + 25 d x

limit as x approaches 27 of (3-\sqrt[3]{x})/(27-x)

x \to 27 lim (\frac{3 - 3 x}{27 - x})

tangent f(x)=(9-6x)^2,(1,9)

t an g e n t f (x) = (9 - 6 x)^{2}, (1, 9)

inverselaplace 1/(s^2(1+7s))

in v erse l a pl a ce \frac{1}{s ^{2} ( 1 + 7 s )}