integral of x^2(32+4x^3)^6

解

\int x^{2} (32 + 4 x^{3})^{6} d x

\frac{3 2 ^{6} x ^{3}}{3} + 134217728 x^{6} + \frac{83886080 x ^{9}}{3} + \frac{10485760 x ^{12}}{3} + 262144 x^{15} + \frac{32768 x ^{18}}{3} + \frac{4096 x ^{21}}{21} + C

解答ステップ

\int x^{2} (32 + 4 x^{3})^{6} d x

拡張

x^{2} (32 + 4 x^{3})^{6} : 3 2^{6} x^{2} + 805306368 x^{5} + 251658240 x^{8} + 41943040 x^{11} + 3932160 x^{14} + 196608 x^{17} + 4096 x^{20}

= \int 3 2^{6} x^{2} + 805306368 x^{5} + 251658240 x^{8} + 41943040 x^{11} + 3932160 x^{14} + 196608 x^{17} + 4096 x^{20} d x

総和規則を適用する:

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

= \int 3 2^{6} x^{2} d x + \int 805306368 x^{5} d x + \int 251658240 x^{8} d x + \int 41943040 x^{11} d x + \int 3932160 x^{14} d x + \int 196608 x^{17} d x + \int 4096 x^{20} d x

\int 3 2^{6} x^{2} d x = \frac{3 2 ^{6} x ^{3}}{3}

\int 805306368 x^{5} d x = 134217728 x^{6}

\int 251658240 x^{8} d x = \frac{83886080 x ^{9}}{3}

\int 41943040 x^{11} d x = \frac{10485760 x ^{12}}{3}

\int 3932160 x^{14} d x = 262144 x^{15}

\int 196608 x^{17} d x = \frac{32768 x ^{18}}{3}

\int 4096 x^{20} d x = \frac{4096 x ^{21}}{21}

= \frac{3 2 ^{6} x ^{3}}{3} + 134217728 x^{6} + \frac{83886080 x ^{9}}{3} + \frac{10485760 x ^{12}}{3} + 262144 x^{15} + \frac{32768 x ^{18}}{3} + \frac{4096 x ^{21}}{21}

定数を解答に追加する

= \frac{3 2 ^{6} x ^{3}}{3} + 134217728 x^{6} + \frac{83886080 x ^{9}}{3} + \frac{10485760 x ^{12}}{3} + 262144 x^{15} + \frac{32768 x ^{18}}{3} + \frac{4096 x ^{21}}{21} + C