integral of (x^2-4)^{3/2}

解

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

x (x^{2} - 4)^{\frac{3}{2}} - 48 - \frac{1}{32} x (x^{2} - 4)^{\frac{1}{2}} + 4 ln \frac{( x ^{2} - 4 ) ^{\frac{1}{2}} + x}{2} + \frac{1}{64} x^{3} (x^{2} - 4)^{\frac{1}{2}} + C

解答ステップ

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

部分積分を適用する

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

\int 3 x^{2} (x^{2} - 4)^{\frac{1}{2}} d x = 48 - \frac{1}{32} x (x^{2} - 4)^{\frac{1}{2}} + 4 ln \frac{( x ^{2} - 4 ) ^{\frac{1}{2}} + x}{2} + \frac{1}{64} x^{3} (x^{2} - 4)^{\frac{1}{2}}

= x (x^{2} - 4)^{\frac{3}{2}} - 48 - \frac{1}{32} x (x^{2} - 4)^{\frac{1}{2}} + 4 ln \frac{( x ^{2} - 4 ) ^{\frac{1}{2}} + x}{2} + \frac{1}{64} x^{3} (x^{2} - 4)^{\frac{1}{2}}

定数を解答に追加する

= x (x^{2} - 4)^{\frac{3}{2}} - 48 - \frac{1}{32} x (x^{2} - 4)^{\frac{1}{2}} + 4 ln \frac{( x ^{2} - 4 ) ^{\frac{1}{2}} + x}{2} + \frac{1}{64} x^{3} (x^{2} - 4)^{\frac{1}{2}} + C