integral of x^2(x^2-25)^{3/2}

解

\int x^{2} (x^{2} - 25)^{\frac{3}{2}} d x

\frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}})) + C

解答ステップ

\int x^{2} (x^{2} - 25)^{\frac{3}{2}} d x

簡素化

x^{2} : xx

= \int xx (x^{2} - 25)^{\frac{3}{2}} d x

部分積分を適用する

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - \int x^{4} (x^{2} - 25)^{\frac{1}{2}} d x

\int x^{4} (x^{2} - 25)^{\frac{1}{2}} d x = 15625 - \frac{1}{2500} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{200} x (x^{2} - 25)^{\frac{1}{2}} + 25 ln \frac{( x ^{2} - 25 ) ^{\frac{1}{2}} + x}{5} + \frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{3000} x^{3} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{80} x (x^{2} - 25)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{5} (x^{2} - 25)^{\frac{1}{2}} + x)

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 - \frac{1}{2500} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{200} x (x^{2} - 25)^{\frac{1}{2}} + 25 ln \frac{( x ^{2} - 25 ) ^{\frac{1}{2}} + x}{5} + \frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{3000} x^{3} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{80} x (x^{2} - 25)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{5} (x^{2} - 25)^{\frac{1}{2}} + x)

簡素化

\frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 - \frac{1}{2500} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{200} x (x^{2} - 25)^{\frac{1}{2}} + 25 ln \frac{( x ^{2} - 25 ) ^{\frac{1}{2}} + x}{5} + \frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{3000} x^{3} (x^{2} - 25)^{\frac{1}{2}} + \frac{1}{80} x (x^{2} - 25)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{5} (x^{2} - 25)^{\frac{1}{2}} + x) : \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}))

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}))

定数を解答に追加する

= \frac{1}{3} x^{3} (x^{2} - 25)^{\frac{3}{2}} - 15625 (\frac{1}{93750} x^{5} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{15000} x^{3} (x^{2} - 25)^{\frac{1}{2}} - \frac{1}{400} x (x^{2} - 25)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{5} x + (x^{2} - 25)^{\frac{1}{2}})) + C