integral of x^3(1+x^4)^3

Lösung

\int x^{3} (1 + x^{4})^{3} d x

\frac{x ^{4}}{4} + \frac{3 x ^{8}}{8} + \frac{x ^{12}}{4} + \frac{x ^{16}}{16} + C

Schritte zur Lösung

\int x^{3} (1 + x^{4})^{3} d x

Multipliziere aus

x^{3} (1 + x^{4})^{3} : x^{3} + 3 x^{7} + 3 x^{11} + x^{15}

= \int x^{3} + 3 x^{7} + 3 x^{11} + x^{15} 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 x^{3} d x + \int 3 x^{7} d x + \int 3 x^{11} d x + \int x^{15} d x

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

\int 3 x^{7} d x = \frac{3 x ^{8}}{8}

\int 3 x^{11} d x = \frac{x ^{12}}{4}

\int x^{15} d x = \frac{x ^{16}}{16}

= \frac{x ^{4}}{4} + \frac{3 x ^{8}}{8} + \frac{x ^{12}}{4} + \frac{x ^{16}}{16}

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{4}}{4} + \frac{3 x ^{8}}{8} + \frac{x ^{12}}{4} + \frac{x ^{16}}{16} + C

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