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人気のある微分積分 >

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

解

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

解

6 (- arctan (\frac{x}{3}) + \frac{x}{3}) + C

解答ステップ

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

定数を除く:

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

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

\frac{x ^{2}}{x ^{2} + 9} = - \frac{9}{x ^{2} + 9} + 1

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

置換積分法を適用する

= 2 \cdot \int 3 (- \frac{1}{u ^{2} + 1} + 1) d u

定数を除く:

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

= 2 \cdot 3 \cdot \int - \frac{1}{u ^{2} + 1} + 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

= 2 \cdot 3 (- \int \frac{1}{u ^{2} + 1} d u + \int 1 d u)

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

\int 1 d u = u

= 2 \cdot 3 (- arctan (u) + u)

代用を戻す

u = \frac{x}{3}

= 2 \cdot 3 (- arctan (\frac{x}{3}) + \frac{x}{3})

簡素化

= 6 (- arctan (\frac{x}{3}) + \frac{x}{3})

定数を解答に追加する

= 6 (- arctan (\frac{x}{3}) + \frac{x}{3}) + C

グラフ

人気の例

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