integral of (5x+3)/(x^2+1)

解

\int \frac{5 x + 3}{x ^{2} + 1} d x

\frac{5}{2} ln x^{2} + 1 + 3 arctan (x) + C

解答ステップ

\int \frac{5 x + 3}{x ^{2} + 1} d x

拡張

\frac{5 x + 3}{x ^{2} + 1} : \frac{5 x}{x ^{2} + 1} + \frac{3}{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{5 x}{x ^{2} + 1} d x + \int \frac{3}{x ^{2} + 1} d x

\int \frac{5 x}{x ^{2} + 1} d x = \frac{5}{2} ln x^{2} + 1

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

= \frac{5}{2} ln x^{2} + 1 + 3 arctan (x)

定数を解答に追加する

= \frac{5}{2} ln x^{2} + 1 + 3 arctan (x) + C