integral of (4+9x)/(1+x^2)

解

\int \frac{4 + 9 x}{1 + x ^{2}} d x

4 arctan (x) + \frac{9}{2} ln 1 + x^{2} + C

解答ステップ

\int \frac{4 + 9 x}{1 + x ^{2}} d x

拡張

\frac{4 + 9 x}{1 + x ^{2}} : \frac{4}{1 + x ^{2}} + \frac{9 x}{1 + x ^{2}}

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}{1 + x ^{2}} d x + \int \frac{9 x}{1 + x ^{2}} d x

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

\int \frac{9 x}{1 + x ^{2}} d x = \frac{9}{2} ln 1 + x^{2}

= 4 arctan (x) + \frac{9}{2} ln 1 + x^{2}

定数を解答に追加する

= 4 arctan (x) + \frac{9}{2} ln 1 + x^{2} + C