integral of (2+sin(x))/(sqrt(x))

解

\int \frac{2 + sin ( x )}{x} d x

2 (2 x + \frac{π}{2} § (\frac{2}{π} x)) + C

解答ステップ

\int \frac{2 + sin ( x )}{x} d x

u置換積分法を適用する

= \int 2 (2 + sin (u^{2})) d u

定数を除く:

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

= 2 \cdot \int 2 + sin (u^{2}) d u

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

= 2 (\int 2 d u + \int sin (u^{2}) d u)

\int 2 d u = 2 u

\int sin (u^{2}) d u = \frac{π}{2} § (\frac{2}{π} u)

= 2 (2 u + \frac{π}{2} § (\frac{2}{π} u))

代用を戻す

u = x

= 2 (2 x + \frac{π}{2} § (\frac{2}{π} x))

定数を解答に追加する

= 2 (2 x + \frac{π}{2} § (\frac{2}{π} x)) + C