integral of 1/(sqrt((x-8)(6-x)))

解

\int \frac{1}{( x - 8 ) ( 6 - x )} d x

arcsin (x - 7) + C

解答ステップ

\int \frac{1}{( x - 8 ) ( 6 - x )} d x

\frac{1}{( x - 8 ) ( 6 - x )} = \frac{1 - ( x - 8 ) ( 6 - x )}{( x - 8 ) ( 6 - x )} + 1

= \int \frac{1 - ( x - 8 ) ( 6 - x )}{( x - 8 ) ( 6 - x )} + 1 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{1 - ( x - 8 ) ( 6 - x )}{( x - 8 ) ( 6 - x )} d x + \int 1 d x

\int \frac{1 - ( x - 8 ) ( 6 - x )}{( x - 8 ) ( 6 - x )} d x = arcsin (x - 7) - x

\int 1 d x = x

= arcsin (x - 7) - x + x

簡素化

= arcsin (x - 7)

定数を解答に追加する

= arcsin (x - 7) + C