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人気のある微分積分 >

integral of x/(x^4+4)

解

\int \frac{x}{x ^{4} + 4} d x

解

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

解答ステップ

\int \frac{x}{x ^{4} + 4} d x

u置換積分法を適用する

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

定数を除く:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{2} + 4} d u

置換積分法を適用する

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

定数を除く:

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

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{2} + 1} d v

共通積分を使用する:

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

= \frac{1}{2} \cdot \frac{1}{2} arctan (v)

代用を戻す

= \frac{1}{2} \cdot \frac{1}{2} arctan (\frac{x ^{2}}{2})

簡素化

\frac{1}{2} \cdot \frac{1}{2} arctan (\frac{x ^{2}}{2}) : \frac{1}{4} arctan (\frac{x ^{2}}{2})

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

定数を解答に追加する

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

グラフ

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