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人気のある微分積分 >

integral of (5x)/((x^2+1)(x-2))

解

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

解

- ln x^{2} + 1 + arctan (x) + 2 ln ∣ x - 2 ∣ + C

解答ステップ

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

定数を除く:

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

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

以下の部分分数を得る:

\frac{x}{( x ^{2} + 1 ) ( x - 2 )} : \frac{- 2 x + 1}{5 ( x ^{2} + 1 )} + \frac{2}{5 ( x - 2 )}

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

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

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

\int \frac{- 2 x + 1}{5 ( x ^{2} + 1 )} d x = \frac{1}{5} (- ln x^{2} + 1 + arctan (x))

\int \frac{2}{5 ( x - 2 )} d x = \frac{2}{5} ln ∣ x - 2 ∣

= 5 (\frac{1}{5} (- ln x^{2} + 1 + arctan (x)) + \frac{2}{5} ln ∣ x - 2 ∣)

簡素化

5 (\frac{1}{5} (- ln x^{2} + 1 + arctan (x)) + \frac{2}{5} ln ∣ x - 2 ∣) : - ln x^{2} + 1 + arctan (x) + 2 ln ∣ x - 2 ∣

= - ln x^{2} + 1 + arctan (x) + 2 ln ∣ x - 2 ∣

定数を解答に追加する

= - ln x^{2} + 1 + arctan (x) + 2 ln ∣ x - 2 ∣ + C

グラフ

人気の例

integral of (e^{9/x})/(x^2)

\int \frac{e ^{\frac{9}{x}}}{x ^{2}} d x

integral of 64xsec^2(4x)

\int 64 x sec^{2} (4 x) d x

integral of z/((a^2+z^2)^{3/2)}

\int \frac{z}{( a ^{2} + z ^{2} ) ^{\frac{3}{2}}} d z

integral of ((5x^2+10x-30))/(x^2+x-6)

\int \frac{( 5 x ^{2} + 10 x - 30 )}{x ^{2} + x - 6} d x

integral of (16x)/(x^2+1)

\int \frac{16 x}{x ^{2} + 1} d x