integral of e^{11x}(8x^3-2x^2-24x+1)

Lösung

\int e^{11 x} (8 x^{3} - 2 x^{2} - 24 x + 1) d x

\frac{8}{11} e^{11 x} x^{3} - \frac{46}{121} e^{11 x} x^{2} - \frac{2812}{1331} e^{11 x} x + \frac{4143}{14641} e^{11 x} + C

Schritte zur Lösung

\int e^{11 x} (8 x^{3} - 2 x^{2} - 24 x + 1) d x

Multipliziere aus

e^{11 x} (8 x^{3} - 2 x^{2} - 24 x + 1) : 8 e^{11 x} x^{3} - 2 e^{11 x} x^{2} - 24 e^{11 x} x + e^{11 x}

= \int 8 e^{11 x} x^{3} - 2 e^{11 x} x^{2} - 24 e^{11 x} x + e^{11 x} d x

Wende die Summenregel an:

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

= \int 8 e^{11 x} x^{3} d x - \int 2 e^{11 x} x^{2} d x - \int 24 e^{11 x} x d x + \int e^{11 x} d x

\int 8 e^{11 x} x^{3} d x = \frac{8}{14641} (1331 e^{11 x} x^{3} - 3 (121 e^{11 x} x^{2} - 2 (11 e^{11 x} x - e^{11 x})))

\int 2 e^{11 x} x^{2} d x = 2 (\frac{1}{11} e^{11 x} x^{2} - \frac{2}{1331} (11 e^{11 x} x - e^{11 x}))

\int 24 e^{11 x} x d x = \frac{24}{121} (11 e^{11 x} x - e^{11 x})

\int e^{11 x} d x = \frac{1}{11} e^{11 x}

= \frac{8}{14641} (1331 e^{11 x} x^{3} - 3 (121 e^{11 x} x^{2} - 2 (11 e^{11 x} x - e^{11 x}))) - 2 (\frac{1}{11} e^{11 x} x^{2} - \frac{2}{1331} (11 e^{11 x} x - e^{11 x})) - \frac{24}{121} (11 e^{11 x} x - e^{11 x}) + \frac{1}{11} e^{11 x}

Vereinfache

\frac{8}{14641} (1331 e^{11 x} x^{3} - 3 (121 e^{11 x} x^{2} - 2 (11 e^{11 x} x - e^{11 x}))) - 2 (\frac{1}{11} e^{11 x} x^{2} - \frac{2}{1331} (11 e^{11 x} x - e^{11 x})) - \frac{24}{121} (11 e^{11 x} x - e^{11 x}) + \frac{1}{11} e^{11 x} : \frac{8}{11} e^{11 x} x^{3} - \frac{46}{121} e^{11 x} x^{2} - \frac{2812}{1331} e^{11 x} x + \frac{4143}{14641} e^{11 x}

= \frac{8}{11} e^{11 x} x^{3} - \frac{46}{121} e^{11 x} x^{2} - \frac{2812}{1331} e^{11 x} x + \frac{4143}{14641} e^{11 x}

Füge eine Konstante zur Lösung hinzu

= \frac{8}{11} e^{11 x} x^{3} - \frac{46}{121} e^{11 x} x^{2} - \frac{2812}{1331} e^{11 x} x + \frac{4143}{14641} e^{11 x} + C

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