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人気のある微分積分 >

integral of (5x+2)/(sqrt(x^2+2x+10))

解

\int \frac{5 x + 2}{x ^{2} + 2 x + 10} d x

解

5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10) + C

解答ステップ

\int \frac{5 x + 2}{x ^{2} + 2 x + 10} d x

平方完成する

x^{2} + 2 x + 10 : (x + 1)^{2} + 9

= \int \frac{5 x + 2}{( x + 1 ) ^{2} + 9} d x

u置換積分法を適用する

= \int \frac{5 u - 3}{u ^{2} + 9} d u

拡張

\frac{5 u - 3}{u ^{2} + 9} : \frac{5 u}{u ^{2} + 9} - \frac{3}{u ^{2} + 9}

= \int \frac{5 u}{u ^{2} + 9} - \frac{3}{u ^{2} + 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

= \int \frac{5 u}{u ^{2} + 9} d u - \int \frac{3}{u ^{2} + 9} d u

\int \frac{5 u}{u ^{2} + 9} d u = 5 u^{2} + 9

\int \frac{3}{u ^{2} + 9} d u = 3 ln (\frac{1}{3} u + 9 + u^{2})

= 5 u^{2} + 9 - 3 ln (\frac{1}{3} u + 9 + u^{2})

代用を戻す

u = x + 1

= 5 (x + 1)^{2} + 9 - 3 ln (\frac{1}{3} x + 1 + 9 + (x + 1)^{2})

簡素化

5 (x + 1)^{2} + 9 - 3 ln (\frac{1}{3} x + 1 + 9 + (x + 1)^{2}) : 5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10)

= 5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10)

定数を解答に追加する

= 5 x^{2} + 2 x + 10 - 3 ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10) + C

グラフ

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