integral of (x^2+45)/(x^2+49)

解

\int \frac{x ^{2} + 45}{x ^{2} + 49} d x

- \frac{4}{7} arctan (\frac{x}{7}) + x + C

解答ステップ

\int \frac{x ^{2} + 45}{x ^{2} + 49} d x

\frac{x ^{2} + 45}{x ^{2} + 49} = - \frac{4}{x ^{2} + 49} + 1

= \int - \frac{4}{x ^{2} + 49} + 1 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{4}{x ^{2} + 49} d x + \int 1 d x

\int \frac{4}{x ^{2} + 49} d x = \frac{4}{7} arctan (\frac{x}{7})

\int 1 d x = x

= - \frac{4}{7} arctan (\frac{x}{7}) + x

定数を解答に追加する

= - \frac{4}{7} arctan (\frac{x}{7}) + x + C