integral of (x^2+3x+1)/(x(x^2+1))

解

\int \frac{x ^{2} + 3 x + 1}{x ( x ^{2} + 1 )} d x

3 arctan (x) + ln ∣ x ∣ + C

解答ステップ

\int \frac{x ^{2} + 3 x + 1}{x ( x ^{2} + 1 )} d x

拡張

\frac{x ^{2} + 3 x + 1}{x ( x ^{2} + 1 )} : \frac{x ^{2}}{x ( x ^{2} + 1 )} + \frac{3 x}{x ( x ^{2} + 1 )} + \frac{1}{x ( x ^{2} + 1 )}

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{x ^{2}}{x ( x ^{2} + 1 )} d x + \int \frac{3 x}{x ( x ^{2} + 1 )} d x + \int \frac{1}{x ( x ^{2} + 1 )} d x

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

\int \frac{3 x}{x ( x ^{2} + 1 )} d x = 3 arctan (x)

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

= \frac{1}{2} ln x^{2} + 1 + 3 arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

簡素化

\frac{1}{2} ln x^{2} + 1 + 3 arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 : 3 arctan (x) + ln ∣ x ∣

= 3 arctan (x) + ln ∣ x ∣

定数を解答に追加する

= 3 arctan (x) + ln ∣ x ∣ + C