integral of (3x+4)^{5/2}

解

\int (3 x + 4)^{\frac{5}{2}} d x

\frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}} + C

解答ステップ

\int (3 x + 4)^{\frac{5}{2}} d x

部分積分を適用する

= x (3 x + 4)^{\frac{5}{2}} - \int \frac{15}{2} x (3 x + 4)^{\frac{3}{2}} d x

\int \frac{15}{2} x (3 x + 4)^{\frac{3}{2}} d x = \frac{1}{21} (15 x - 8) (3 x + 4)^{\frac{5}{2}}

= x (3 x + 4)^{\frac{5}{2}} - \frac{1}{21} (15 x - 8) (3 x + 4)^{\frac{5}{2}}

簡素化

x (3 x + 4)^{\frac{5}{2}} - \frac{1}{21} (15 x - 8) (3 x + 4)^{\frac{5}{2}} : \frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}}

= \frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}}

定数を解答に追加する

= \frac{2}{7} x (3 x + 4)^{\frac{5}{2}} + \frac{8}{21} (3 x + 4)^{\frac{5}{2}} + C