integral of (6-72x)/(sqrt(1-36x^2))

解

\int \frac{6 - 72 x}{1 - 36 x ^{2}} d x

arcsin (6 x) + 2 1 - 36 x^{2} + C

解答ステップ

\int \frac{6 - 72 x}{1 - 36 x ^{2}} d x

拡張

\frac{6 - 72 x}{1 - 36 x ^{2}} : \frac{6}{1 - 36 x ^{2}} - \frac{72 x}{1 - 36 x ^{2}}

= \int \frac{6}{1 - 36 x ^{2}} - \frac{72 x}{1 - 36 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{6}{1 - 36 x ^{2}} d x - \int \frac{72 x}{1 - 36 x ^{2}} d x

\int \frac{6}{1 - 36 x ^{2}} d x = arcsin (6 x)

\int \frac{72 x}{1 - 36 x ^{2}} d x = - 2 1 - 36 x^{2}

= arcsin (6 x) - (- 2 1 - 36 x^{2})

簡素化

= arcsin (6 x) + 2 1 - 36 x^{2}

定数を解答に追加する

= arcsin (6 x) + 2 1 - 36 x^{2} + C