integral of (5x+3)/(x^2+4)

解

\int \frac{5 x + 3}{x ^{2} + 4} d x

\frac{5}{2} ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + C

解答ステップ

\int \frac{5 x + 3}{x ^{2} + 4} d x

拡張

\frac{5 x + 3}{x ^{2} + 4} : \frac{5 x}{x ^{2} + 4} + \frac{3}{x ^{2} + 4}

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

\int \frac{5 x}{x ^{2} + 4} d x = \frac{5}{2} ln x^{2} + 4

\int \frac{3}{x ^{2} + 4} d x = \frac{3}{2} arctan (\frac{x}{2})

= \frac{5}{2} ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2})

定数を解答に追加する

= \frac{5}{2} ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + C