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人気のある微分積分 >

integral of (x^2+3)/(xsqrt(x^2-9))

解

\int \frac{x ^{2} + 3}{x x ^{2} - 9} d x

解

x^{2} - 9 + arctan (\frac{1}{3} x^{2} - 9) + C

解答ステップ

\int \frac{x ^{2} + 3}{x x ^{2} - 9} d x

u置換積分法を適用する

= \int \frac{u + 3}{2 u u - 9} 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{u + 3}{u u - 9} d u

拡張

\frac{u + 3}{u u - 9} : \frac{1}{u - 9} + \frac{3}{u u - 9}

= \frac{1}{2} \cdot \int \frac{1}{u - 9} + \frac{3}{u u - 9} 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}{2} (\int \frac{1}{u - 9} d u + \int \frac{3}{u u - 9} d u)

\int \frac{1}{u - 9} d u = 2 u - 9

\int \frac{3}{u u - 9} d u = 2 arctan (\frac{1}{3} u - 9)

= \frac{1}{2} (2 u - 9 + 2 arctan (\frac{1}{3} u - 9))

代用を戻す

u = x^{2}

= \frac{1}{2} (2 x^{2} - 9 + 2 arctan (\frac{1}{3} x^{2} - 9))

簡素化

\frac{1}{2} (2 x^{2} - 9 + 2 arctan (\frac{1}{3} x^{2} - 9)) : x^{2} - 9 + arctan (\frac{1}{3} x^{2} - 9)

= x^{2} - 9 + arctan (\frac{1}{3} x^{2} - 9)

定数を解答に追加する

= x^{2} - 9 + arctan (\frac{1}{3} x^{2} - 9) + C

グラフ

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