integral of x^2(x^2-49)^{3/2}

解答

\int x^{2} (x^{2} - 49)^{\frac{3}{2}} d x

\frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - 117649 (\frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{784} x (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}})) + C

求解步骤

\int x^{2} (x^{2} - 49)^{\frac{3}{2}} d x

化简

x^{2} : xx

= \int xx (x^{2} - 49)^{\frac{3}{2}} d x

使用分布积分法

= \frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - \int x^{4} (x^{2} - 49)^{\frac{1}{2}} d x

\int x^{4} (x^{2} - 49)^{\frac{1}{2}} d x = 117649 - \frac{1}{9604} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{392} x (x^{2} - 49)^{\frac{1}{2}} + 49 ln \frac{( x ^{2} - 49 ) ^{\frac{1}{2}} + x}{7} + \frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{784} x (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{7} (x^{2} - 49)^{\frac{1}{2}} + x)

= \frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - 117649 - \frac{1}{9604} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{392} x (x^{2} - 49)^{\frac{1}{2}} + 49 ln \frac{( x ^{2} - 49 ) ^{\frac{1}{2}} + x}{7} + \frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{784} x (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{7} (x^{2} - 49)^{\frac{1}{2}} + x)

化简

\frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - 117649 - \frac{1}{9604} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{392} x (x^{2} - 49)^{\frac{1}{2}} + 49 ln \frac{( x ^{2} - 49 ) ^{\frac{1}{2}} + x}{7} + \frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{784} x (x^{2} - 49)^{\frac{1}{2}} + \frac{5}{16} ln (\frac{1}{7} (x^{2} - 49)^{\frac{1}{2}} + x) : \frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - 117649 (\frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{784} x (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}}))

= \frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - 117649 (\frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{784} x (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}}))

解答补常数

= \frac{1}{3} x^{3} (x^{2} - 49)^{\frac{3}{2}} - 117649 (\frac{1}{705894} x^{5} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{57624} x^{3} (x^{2} - 49)^{\frac{1}{2}} - \frac{1}{784} x (x^{2} - 49)^{\frac{1}{2}} - \frac{3}{8} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}}) + \frac{5}{16} ln (\frac{1}{7} x + (x^{2} - 49)^{\frac{1}{2}})) + C

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