integral of (12x^3-3/(x^3)+5e^{-x})

Lösung

\int (12 x^{3} - \frac{3}{x ^{3}} + 5 e^{- x}) d x

3 x^{4} + \frac{3}{2 x ^{2}} - 5 e^{- x} + C

Schritte zur Lösung

\int 12 x^{3} - \frac{3}{x ^{3}} + 5 e^{- 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 12 x^{3} d x - \int \frac{3}{x ^{3}} d x + \int 5 e^{- x} d x

\int 12 x^{3} d x = 3 x^{4}

\int \frac{3}{x ^{3}} d x = - \frac{3}{2 x ^{2}}

\int 5 e^{- x} d x = - 5 e^{- x}

= 3 x^{4} - (- \frac{3}{2 x ^{2}}) - 5 e^{- x}

Vereinfache

= 3 x^{4} + \frac{3}{2 x ^{2}} - 5 e^{- x}

Füge eine Konstante zur Lösung hinzu

= 3 x^{4} + \frac{3}{2 x ^{2}} - 5 e^{- x} + C

\int \frac{3 x + 27}{x} d x

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