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Beliebt Rechnen >

integral of 4x^3(2x^4+4)^7

Lösung

\int 4 x^{3} (2 x^{4} + 4)^{7} d x

Lösung

4 (4 x^{32} + 64 x^{28} + 448 x^{24} + 1792 x^{20} + 4480 x^{16} + 7168 x^{12} + 7168 x^{8} + 4096 x^{4}) + C

Schritte zur Lösung

\int 4 x^{3} (2 x^{4} + 4)^{7} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int x^{3} (2 x^{4} + 4)^{7} d x

Multipliziere aus

x^{3} (2 x^{4} + 4)^{7} : 128 x^{31} + 1792 x^{27} + 10752 x^{23} + 35840 x^{19} + 71680 x^{15} + 86016 x^{11} + 57344 x^{7} + 16384 x^{3}

= 4 \cdot \int 128 x^{31} + 1792 x^{27} + 10752 x^{23} + 35840 x^{19} + 71680 x^{15} + 86016 x^{11} + 57344 x^{7} + 16384 x^{3} 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

= 4 (\int 128 x^{31} d x + \int 1792 x^{27} d x + \int 10752 x^{23} d x + \int 35840 x^{19} d x + \int 71680 x^{15} d x + \int 86016 x^{11} d x + \int 57344 x^{7} d x + \int 16384 x^{3} d x)

\int 128 x^{31} d x = 4 x^{32}

\int 1792 x^{27} d x = 64 x^{28}

\int 10752 x^{23} d x = 448 x^{24}

\int 35840 x^{19} d x = 1792 x^{20}

\int 71680 x^{15} d x = 4480 x^{16}

\int 86016 x^{11} d x = 7168 x^{12}

\int 57344 x^{7} d x = 7168 x^{8}

\int 16384 x^{3} d x = 4096 x^{4}

= 4 (4 x^{32} + 64 x^{28} + 448 x^{24} + 1792 x^{20} + 4480 x^{16} + 7168 x^{12} + 7168 x^{8} + 4096 x^{4})

Füge eine Konstante zur Lösung hinzu

= 4 (4 x^{32} + 64 x^{28} + 448 x^{24} + 1792 x^{20} + 4480 x^{16} + 7168 x^{12} + 7168 x^{8} + 4096 x^{4}) + C

Graph

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