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人気のある微分積分 >

integral of (x^2)/(sqrt(a^2+(x-b)^2))

解

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

解

\frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + C

解答ステップ

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

u置換積分法を適用する

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

拡張

\frac{( u + b ) ^{2}}{a ^{2} + u ^{2}} : \frac{u ^{2}}{a ^{2} + u ^{2}} + \frac{2 b u}{a ^{2} + u ^{2}} + \frac{b ^{2}}{a ^{2} + u ^{2}}

= \int \frac{u ^{2}}{a ^{2} + u ^{2}} + \frac{2 b u}{a ^{2} + u ^{2}} + \frac{b ^{2}}{a ^{2} + u ^{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

= \int \frac{u ^{2}}{a ^{2} + u ^{2}} d u + \int \frac{2 b u}{a ^{2} + u ^{2}} d u + \int \frac{b ^{2}}{a ^{2} + u ^{2}} d u

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

\int \frac{2 b u}{a ^{2} + u ^{2}} d u = 2 b a^{2} + u^{2}

\int \frac{b ^{2}}{a ^{2} + u ^{2}} d u = b^{2} ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

= \frac{1}{2} u u^{2} + a^{2} - \frac{a ^{2}}{2} ln (\frac{u + u ^{2} + a ^{2}}{∣ a ∣}) + 2 b a^{2} + u^{2} + b^{2} ln (\frac{u + a ^{2} + u ^{2}}{∣ a ∣})

代用を戻す

u = x - b

= \frac{1}{2} (x - b) (x - b)^{2} + a^{2} - \frac{a ^{2}}{2} ln \frac{x - b + ( x - b ) ^{2} + a ^{2}}{∣ a ∣} + 2 b a^{2} + (x - b)^{2} + b^{2} ln \frac{x - b + a ^{2} + ( x - b ) ^{2}}{∣ a ∣}

簡素化

\frac{1}{2} (x - b) (x - b)^{2} + a^{2} - \frac{a ^{2}}{2} ln \frac{x - b + ( x - b ) ^{2} + a ^{2}}{∣ a ∣} + 2 b a^{2} + (x - b)^{2} + b^{2} ln \frac{x - b + a ^{2} + ( x - b ) ^{2}}{∣ a ∣} : \frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣}

= \frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣}

定数を解答に追加する

= \frac{1}{2} x a^{2} + (x - b)^{2} + b^{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + \frac{3}{2} b a^{2} + (x - b)^{2} - \frac{a ^{2}}{2} ln \frac{x + a ^{2} + ( x - b ) ^{2} - b}{∣ a ∣} + C

グラフ

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