integral of (5x^2-6)^{3/2}

解

\int (5 x^{2} - 6)^{\frac{3}{2}} d x

x (5 x^{2} - 6)^{\frac{3}{2}} - 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}}) + C

解答ステップ

\int (5 x^{2} - 6)^{\frac{3}{2}} d x

部分積分を適用する

= x (5 x^{2} - 6)^{\frac{3}{2}} - \int 15 x^{2} (5 x^{2} - 6)^{\frac{1}{2}} d x

\int 15 x^{2} (5 x^{2} - 6)^{\frac{1}{2}} d x = 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}})

= x (5 x^{2} - 6)^{\frac{3}{2}} - 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}})

定数を解答に追加する

= x (5 x^{2} - 6)^{\frac{3}{2}} - 15 \cdot 6^{\frac{1}{2}} (\frac{6}{5})^{\frac{3}{2}} (- \frac{1}{48} (5^{\frac{1}{2}} x (5 x^{2} - 6)^{\frac{1}{2}} + 6 ln (\frac{5 x ^{2} - 6}{6})^{\frac{1}{2}} + (\frac{5}{6})^{\frac{1}{2}} x) + \frac{5 \cdot 5 ^{\frac{1}{2}}}{144} x^{3} (5 x^{2} - 6)^{\frac{1}{2}}) + C