integral of (7arcsin(x))/(x^2)

解

\int \frac{7 arcsin ( x )}{x ^{2}} d x

7 (- \frac{arcsin ( x )}{x} - \frac{1}{2} ln 1 - x^{2} + 1 + \frac{1}{2} ln 1 - x^{2} - 1) + C

解答ステップ

\int \frac{7 arcsin ( x )}{x ^{2}} d x

定数を除く:

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

= 7 \cdot \int \frac{arcsin ( x )}{x ^{2}} d x

部分積分を適用する

= 7 (- \frac{arcsin ( x )}{x} - \int - \frac{1}{x 1 - x ^{2}} d x)

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

= 7 (- \frac{arcsin ( x )}{x} - (\frac{1}{2} ln 1 - x^{2} + 1 - \frac{1}{2} ln 1 - x^{2} - 1))

簡素化

= 7 (- \frac{arcsin ( x )}{x} - \frac{1}{2} ln 1 - x^{2} + 1 + \frac{1}{2} ln 1 - x^{2} - 1)

定数を解答に追加する

= 7 (- \frac{arcsin ( x )}{x} - \frac{1}{2} ln 1 - x^{2} + 1 + \frac{1}{2} ln 1 - x^{2} - 1) + C