integral from 1 to e^{-x^2 of}(ln(t))^4

Lösung

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

x^{8} e^{- x^{2}} - 4 (- x^{6} e^{- x^{2}} - 3 (x^{4} e^{- x^{2}} - 2 (- x^{2} e^{- x^{2}} - e^{- x^{2}}))) - 24

Schritte zur Lösung

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

Vereinfache

= \int_{1}^{e^{- x^{2}}} ln^{4} (t) d t

Wende die partielle Integration an

= [t ln^{4} (t) - \int 4 ln^{3} (t) d t]_{1}^{e^{- x^{2}}}

\int 4 ln^{3} (t) d t = 4 (t ln^{3} (t) - 3 (t ln^{2} (t) - 2 (t ln (t) - t)))

= [t ln^{4} (t) - 4 (t ln^{3} (t) - 3 (t ln^{2} (t) - 2 (t ln (t) - t)))]_{1}^{e^{- x^{2}}}

Berechne die Grenzen:

x^{8} e^{- x^{2}} - 4 (- x^{6} e^{- x^{2}} - 3 (x^{4} e^{- x^{2}} - 2 (- x^{2} e^{- x^{2}} - e^{- x^{2}}))) - 24

= x^{8} e^{- x^{2}} - 4 (- x^{6} e^{- x^{2}} - 3 (x^{4} e^{- x^{2}} - 2 (- x^{2} e^{- x^{2}} - e^{- x^{2}}))) - 24

\int_{0}^{\frac{1}{5}} \frac{1}{100 x ^{2} + 4} d x

\int_{1}^{4} e^{b x - b} d x

\int_{1}^{8} 3 x^{2} d x

\int_{0}^{T} (x^{4} - 8 x + 7) d x

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