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

解

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

arctan (x) + \frac{arctan ^{2} ( x )}{2} + C

解答ステップ

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

拡張

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

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

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

= arctan (x) + \frac{arctan ^{2} ( x )}{2}

定数を解答に追加する

= arctan (x) + \frac{arctan ^{2} ( x )}{2} + C