integral of 2/x+1/(4x^2+1)-x^{-3/2}

解

\int \frac{2}{x} + \frac{1}{4 x ^{2} + 1} - x^{- \frac{3}{2}} d x

2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) + \frac{2}{x ^{\frac{1}{2}}} + C

解答ステップ

\int \frac{2}{x} + \frac{1}{4 x ^{2} + 1} - x^{- \frac{3}{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{2}{x} d x + \int \frac{1}{4 x ^{2} + 1} d x - \int x^{- \frac{3}{2}} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{1}{4 x ^{2} + 1} d x = \frac{1}{2} arctan (2 x)

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

= 2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) - (- \frac{2}{x ^{\frac{1}{2}}})

簡素化

= 2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) + \frac{2}{x ^{\frac{1}{2}}}

定数を解答に追加する

= 2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) + \frac{2}{x ^{\frac{1}{2}}} + C