integral of ((x^2+x)^2)/(sqrt(x-1))

解

\int \frac{( x ^{2} + x ) ^{2}}{x - 1} d x

\frac{2}{9} (x - 1)^{\frac{9}{2}} + \frac{12}{7} (x - 1)^{\frac{7}{2}} + 8 (x - 1)^{\frac{3}{2}} + \frac{26}{5} (x - 1)^{\frac{5}{2}} + 8 x - 1 + C

解答ステップ

\int \frac{( x ^{2} + x ) ^{2}}{x - 1} d x

拡張

\frac{( x ^{2} + x ) ^{2}}{x - 1} : \frac{x ^{4}}{x - 1} + \frac{2 x ^{3}}{x - 1} + \frac{x ^{2}}{x - 1}

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

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x ^{4}}{x - 1} d x = 2 (\frac{1}{9} (x - 1)^{\frac{9}{2}} + \frac{4}{7} (x - 1)^{\frac{7}{2}} + \frac{6}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + x - 1)

\int \frac{2 x ^{3}}{x - 1} d x = 4 (\frac{1}{7} (x - 1)^{\frac{7}{2}} + \frac{3}{5} (x - 1)^{\frac{5}{2}} + (x - 1)^{\frac{3}{2}} + x - 1)

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

= 2 (\frac{1}{9} (x - 1)^{\frac{9}{2}} + \frac{4}{7} (x - 1)^{\frac{7}{2}} + \frac{6}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + x - 1) + 4 (\frac{1}{7} (x - 1)^{\frac{7}{2}} + \frac{3}{5} (x - 1)^{\frac{5}{2}} + (x - 1)^{\frac{3}{2}} + x - 1) + \frac{2}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + 2 x - 1

簡素化

2 (\frac{1}{9} (x - 1)^{\frac{9}{2}} + \frac{4}{7} (x - 1)^{\frac{7}{2}} + \frac{6}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + x - 1) + 4 (\frac{1}{7} (x - 1)^{\frac{7}{2}} + \frac{3}{5} (x - 1)^{\frac{5}{2}} + (x - 1)^{\frac{3}{2}} + x - 1) + \frac{2}{5} (x - 1)^{\frac{5}{2}} + \frac{4}{3} (x - 1)^{\frac{3}{2}} + 2 x - 1 : \frac{2}{9} (x - 1)^{\frac{9}{2}} + \frac{12}{7} (x - 1)^{\frac{7}{2}} + 8 (x - 1)^{\frac{3}{2}} + \frac{26}{5} (x - 1)^{\frac{5}{2}} + 8 x - 1

= \frac{2}{9} (x - 1)^{\frac{9}{2}} + \frac{12}{7} (x - 1)^{\frac{7}{2}} + 8 (x - 1)^{\frac{3}{2}} + \frac{26}{5} (x - 1)^{\frac{5}{2}} + 8 x - 1

定数を解答に追加する

= \frac{2}{9} (x - 1)^{\frac{9}{2}} + \frac{12}{7} (x - 1)^{\frac{7}{2}} + 8 (x - 1)^{\frac{3}{2}} + \frac{26}{5} (x - 1)^{\frac{5}{2}} + 8 x - 1 + C