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人気のある微分積分 >

integral of (4x^3+11)/((x^2+2)^2)

解

\int \frac{4 x ^{3} + 11}{( x ^{2} + 2 ) ^{2}} d x

解

2 ln \frac{x ^{2}}{2} + 1 + \frac{4}{x ^{2} + 2} + \frac{11}{4 2} arctan (\frac{x}{2}) + \frac{11 x}{4 ( x ^{2} + 2 )} + C

解答ステップ

\int \frac{4 x ^{3} + 11}{( x ^{2} + 2 ) ^{2}} d x

置換積分法を適用する

= \int \frac{2 ( 8 2 u ^{3} + 11 )}{( 2 u ^{2} + 2 ) ^{2}} d u

定数を除く:

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

= 2 \cdot \int \frac{8 2 u ^{3} + 11}{( 2 u ^{2} + 2 ) ^{2}} d u

以下の部分分数を得る:

\frac{8 2 u ^{3} + 11}{( 2 u ^{2} + 2 ) ^{2}} : \frac{2 2 u}{u ^{2} + 1} + \frac{- 8 2 u + 11}{4 ( u ^{2} + 1 ) ^{2}}

= 2 \cdot \int \frac{2 2 u}{u ^{2} + 1} + \frac{- 8 2 u + 11}{4 ( 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

= 2 (\int \frac{2 2 u}{u ^{2} + 1} d u + \int \frac{- 8 2 u + 11}{4 ( u ^{2} + 1 ) ^{2}} d u)

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

\int \frac{- 8 2 u + 11}{4 ( u ^{2} + 1 ) ^{2}} d u = \frac{2}{u ^{2} + 1} + \frac{11}{8} arctan (u) + \frac{11}{8} \cdot \frac{u}{1 + u ^{2}}

= 2 (2 ln u^{2} + 1 + \frac{2}{u ^{2} + 1} + \frac{11}{8} arctan (u) + \frac{11}{8} \cdot \frac{u}{1 + u ^{2}})

代用を戻す

u = \frac{x}{2}

= 2 2 ln (\frac{x}{2})^{2} + 1 + \frac{2}{( \frac{x}{2} ) ^{2} + 1} + \frac{11}{8} arctan (\frac{x}{2}) + \frac{11}{8} \cdot \frac{\frac{x}{2}}{1 + ( \frac{x}{2} ) ^{2}}

簡素化

2 2 ln (\frac{x}{2})^{2} + 1 + \frac{2}{( \frac{x}{2} ) ^{2} + 1} + \frac{11}{8} arctan (\frac{x}{2}) + \frac{11}{8} \cdot \frac{\frac{x}{2}}{1 + ( \frac{x}{2} ) ^{2}} : 2 ln \frac{x ^{2}}{2} + 1 + \frac{4}{x ^{2} + 2} + \frac{11}{4 2} arctan (\frac{x}{2}) + \frac{11 x}{4 ( x ^{2} + 2 )}

= 2 ln \frac{x ^{2}}{2} + 1 + \frac{4}{x ^{2} + 2} + \frac{11}{4 2} arctan (\frac{x}{2}) + \frac{11 x}{4 ( x ^{2} + 2 )}

定数を解答に追加する

= 2 ln \frac{x ^{2}}{2} + 1 + \frac{4}{x ^{2} + 2} + \frac{11}{4 2} arctan (\frac{x}{2}) + \frac{11 x}{4 ( x ^{2} + 2 )} + C

グラフ

人気の例

integral of (4x^2)/(x+1)

\int \frac{4 x ^{2}}{x + 1} d x

integral of (sqrt(1-x^2))/1

\int \frac{1 - x ^{2}}{1} d x

integral of 1/(x^2)+8x-4

\int \frac{1}{x ^{2}} + 8 x - 4 d x

integral of x^2 1/(x^2+1)

\int x^{2} \frac{1}{x ^{2} + 1} d x

integral of 1/((7x-6)^2)

\int \frac{1}{( 7 x - 6 ) ^{2}} d x