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人気のある微分積分 >

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

解

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

解

\frac{1}{2 a} arctan (\frac{x}{a}) - \frac{x}{2 ( a ^{2} + x ^{2} )} + C

解答ステップ

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

a^{2} + x^{2} = a^{2} (1 + \frac{x ^{2}}{a ^{2}})

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

置換積分法を適用する

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

定数を除く:

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

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

以下の部分分数を得る:

\frac{u ^{2}}{( u ^{2} + 1 ) ^{2}} : \frac{1}{u ^{2} + 1} - \frac{1}{( u ^{2} + 1 ) ^{2}}

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

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

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

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

= \frac{1}{a} (arctan (u) - \frac{1}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

代用を戻す

u = \frac{x}{a}

= \frac{1}{a} (arctan (\frac{x}{a}) - \frac{1}{2} (arctan (\frac{x}{a}) + \frac{\frac{x}{a}}{1 + ( \frac{x}{a} ) ^{2}}))

簡素化

\frac{1}{a} (arctan (\frac{x}{a}) - \frac{1}{2} (arctan (\frac{x}{a}) + \frac{\frac{x}{a}}{1 + ( \frac{x}{a} ) ^{2}})) : \frac{1}{2 a} arctan (\frac{x}{a}) - \frac{x}{2 ( a ^{2} + x ^{2} )}

= \frac{1}{2 a} arctan (\frac{x}{a}) - \frac{x}{2 ( a ^{2} + x ^{2} )}

定数を解答に追加する

= \frac{1}{2 a} arctan (\frac{x}{a}) - \frac{x}{2 ( a ^{2} + x ^{2} )} + C

グラフ

人気の例

積分記号 (2x)/(sqrt(9x^2-6x-8))

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