integral of x^2(x^2-16)^{3/2}

解

\int x^{2} (x^{2} - 16)^{\frac{3}{2}} d x

\frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - 4096 (\frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{256} x (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}})) + C

解答ステップ

\int x^{2} (x^{2} - 16)^{\frac{3}{2}} d x

簡素化

x^{2} : xx

= \int xx (x^{2} - 16)^{\frac{3}{2}} d x

部分積分を適用する

= \frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - \int x^{4} (x^{2} - 16)^{\frac{1}{2}} d x

\int x^{4} (x^{2} - 16)^{\frac{1}{2}} d x = 4096 - \frac{1}{1024} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{128} x (x^{2} - 16)^{\frac{1}{2}} + 16 ln \frac{( x ^{2} - 16 ) ^{\frac{1}{2}} + x}{4} + \frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{256} x (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{4} (x^{2} - 16)^{\frac{1}{2}} + x)

= \frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - 4096 - \frac{1}{1024} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{128} x (x^{2} - 16)^{\frac{1}{2}} + 16 ln \frac{( x ^{2} - 16 ) ^{\frac{1}{2}} + x}{4} + \frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{256} x (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{4} (x^{2} - 16)^{\frac{1}{2}} + x)

簡素化

\frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - 4096 - \frac{1}{1024} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{128} x (x^{2} - 16)^{\frac{1}{2}} + 16 ln \frac{( x ^{2} - 16 ) ^{\frac{1}{2}} + x}{4} + \frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{256} x (x^{2} - 16)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{4} (x^{2} - 16)^{\frac{1}{2}} + x) : \frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - 4096 (\frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{256} x (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}}))

= \frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - 4096 (\frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{256} x (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}}))

定数を解答に追加する

= \frac{1}{3} x^{3} (x^{2} - 16)^{\frac{3}{2}} - 4096 (\frac{1}{24576} x^{5} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{6144} x^{3} (x^{2} - 16)^{\frac{1}{2}} - \frac{1}{256} x (x^{2} - 16)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{4} x + (x^{2} - 16)^{\frac{1}{2}})) + C