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Beliebt Rechnen >

integral of ((x^5-2x^2))/((1+x^3)^{3/2)}

Lösung

\int \frac{( x ^{5} - 2 x ^{2} )}{( 1 + x ^{3} ) ^{\frac{3}{2}}} d x

Lösung

\frac{2 x ^{3} + 4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}} + \frac{4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}} + C

Schritte zur Lösung

\int \frac{x ^{5} - 2 x ^{2}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} d x

Multipliziere aus

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

= \int \frac{x ^{5}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} - \frac{2 x ^{2}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} 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 \frac{x ^{5}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} d x - \int \frac{2 x ^{2}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} d x

\int \frac{x ^{5}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} d x = \frac{2 x ^{3} + 4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}}

\int \frac{2 x ^{2}}{( 1 + x ^{3} ) ^{\frac{3}{2}}} d x = - \frac{4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}}

= \frac{2 x ^{3} + 4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}} - (- \frac{4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}})

Vereinfache

= \frac{2 x ^{3} + 4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}} + \frac{4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}}

Füge eine Konstante zur Lösung hinzu

= \frac{2 x ^{3} + 4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}} + \frac{4}{3 ( 1 + x ^{3} ) ^{\frac{1}{2}}} + C

Graph

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