integral of x^35^{-x}

Lösung

\int x^{3} 5^{- x} d x

- \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})) + C

Schritte zur Lösung

\int x^{3} \cdot 5^{- x} d x

Wende U-Substitution an

= \int 5^{u} u^{3} d u

Wende die partielle Integration an

= \frac{5 ^{u} u ^{3}}{ln ( 5 )} - \int \frac{3 \cdot 5 ^{u} u ^{2}}{ln ( 5 )} d u

\int \frac{3 \cdot 5 ^{u} u ^{2}}{ln ( 5 )} d u = \frac{3}{ln ( 5 )} (\frac{5 ^{u} u ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )}))

= \frac{5 ^{u} u ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{u} u ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )}))

Setze in

u = - x

ein

= \frac{5 ^{- x} ( - x ) ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} ( - x ) ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{- x} ( - x )}{ln ( 5 )} - \frac{5 ^{- x}}{ln ^{2} ( 5 )}))

Vereinfache

\frac{5 ^{- x} ( - x ) ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} ( - x ) ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (\frac{5 ^{- x} ( - x )}{ln ( 5 )} - \frac{5 ^{- x}}{ln ^{2} ( 5 )})) : - \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}))

= - \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}))

Füge eine Konstante zur Lösung hinzu

= - \frac{5 ^{- x} x ^{3}}{ln ( 5 )} - \frac{3}{ln ( 5 )} (\frac{5 ^{- x} x ^{2}}{ln ( 5 )} - \frac{2}{ln ( 5 )} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})) + C

\int_{0}^{1} e^{\frac{2 x}{3}} d x

\frac{d y}{d x} = \frac{( x + sin ( x ) )}{3 y ^{2}}

d er i v a t i v e x^{3} - 2 x^{2} + 4 x + 3

\frac{\partial}{\partial x} (ln (x^{2} + y^{4} + 2))

n = 1 \sum \infty \frac{n}{n ^{2} + 2}