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人気のある微分積分 >

integral of (x^2)/(25+x^2)

解

\int \frac{x ^{2}}{25 + x ^{2}} d x

解

- 5 arctan (\frac{x}{5}) + x + C

解答ステップ

\int \frac{x ^{2}}{25 + x ^{2}} d x

\frac{x ^{2}}{25 + x ^{2}} = - \frac{25}{25 + x ^{2}} + 1

= \int - \frac{25}{25 + x ^{2}} + 1 d x

置換積分法を適用する

= \int 5 (- \frac{1}{1 + u ^{2}} + 1) d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int - \frac{1}{1 + u ^{2}} + 1 d u

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

= 5 (- \int \frac{1}{1 + u ^{2}} d u + \int 1 d u)

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

\int 1 d u = u

= 5 (- arctan (u) + u)

代用を戻す

u = \frac{x}{5}

= 5 (- arctan (\frac{x}{5}) + \frac{x}{5})

簡素化

5 (- arctan (\frac{x}{5}) + \frac{x}{5}) : - 5 arctan (\frac{x}{5}) + x

= - 5 arctan (\frac{x}{5}) + x

定数を解答に追加する

= - 5 arctan (\frac{x}{5}) + x + C

グラフ

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