integral of ((x-1))/(sqrt(81-4x^2))

解

\int \frac{( x - 1 )}{81 - 4 x ^{2}} d x

- \frac{1}{4} 81 - 4 x^{2} - \frac{1}{2} arcsin (\frac{2}{9} x) + C

解答ステップ

\int \frac{x - 1}{81 - 4 x ^{2}} d x

拡張

\frac{x - 1}{81 - 4 x ^{2}} : \frac{x}{81 - 4 x ^{2}} - \frac{1}{81 - 4 x ^{2}}

= \int \frac{x}{81 - 4 x ^{2}} - \frac{1}{81 - 4 x ^{2}} 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}{81 - 4 x ^{2}} d x - \int \frac{1}{81 - 4 x ^{2}} d x

\int \frac{x}{81 - 4 x ^{2}} d x = - \frac{1}{4} 81 - 4 x^{2}

\int \frac{1}{81 - 4 x ^{2}} d x = \frac{1}{2} arcsin (\frac{2}{9} x)

= - \frac{1}{4} 81 - 4 x^{2} - \frac{1}{2} arcsin (\frac{2}{9} x)

定数を解答に追加する

= - \frac{1}{4} 81 - 4 x^{2} - \frac{1}{2} arcsin (\frac{2}{9} x) + C