integral of (x-3)/(sqrt(4-9x^2))

解

\int \frac{x - 3}{4 - 9 x ^{2}} d x

- \frac{1}{9} 4 - 9 x^{2} - arcsin (\frac{3}{2} x) + C

解答ステップ

\int \frac{x - 3}{4 - 9 x ^{2}} d x

拡張

\frac{x - 3}{4 - 9 x ^{2}} : \frac{x}{4 - 9 x ^{2}} - \frac{3}{4 - 9 x ^{2}}

= \int \frac{x}{4 - 9 x ^{2}} - \frac{3}{4 - 9 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}{4 - 9 x ^{2}} d x - \int \frac{3}{4 - 9 x ^{2}} d x

\int \frac{x}{4 - 9 x ^{2}} d x = - \frac{1}{9} 4 - 9 x^{2}

\int \frac{3}{4 - 9 x ^{2}} d x = arcsin (\frac{3}{2} x)

= - \frac{1}{9} 4 - 9 x^{2} - arcsin (\frac{3}{2} x)

定数を解答に追加する

= - \frac{1}{9} 4 - 9 x^{2} - arcsin (\frac{3}{2} x) + C