integral of 3/(sqrt(5-5x^2))

解

\int \frac{3}{5 - 5 x ^{2}} d x

3 (\frac{x}{5 - 5 x ^{2}} - \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}})) + C

解答ステップ

\int \frac{3}{5 - 5 x ^{2}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{1}{5 - 5 x ^{2}} d x

部分積分を適用する

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

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

= 3 (\frac{x}{5 - 5 x ^{2}} - \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}}))

定数を解答に追加する

= 3 (\frac{x}{5 - 5 x ^{2}} - \frac{1}{5} (- arcsin (x) + \frac{x}{1 - x ^{2}})) + C