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人気のある微分積分 >

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

解

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

解

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

解答ステップ

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

以下の部分分数を得る:

\frac{x ^{2}}{( x - 2 ) ( x ^{2} + 1 )} : \frac{4}{5 ( x - 2 )} + \frac{x + 2}{5 ( x ^{2} + 1 )}

= 5 \cdot \int \frac{4}{5 ( x - 2 )} + \frac{x + 2}{5 ( x ^{2} + 1 )} 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{4}{5 ( x - 2 )} d x + \int \frac{x + 2}{5 ( x ^{2} + 1 )} d x)

\int \frac{4}{5 ( x - 2 )} d x = \frac{4}{5} ln ∣ x - 2 ∣

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

= 5 (\frac{4}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)))

簡素化

5 (\frac{4}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x))) : 4 ln ∣ x - 2 ∣ + \frac{1}{2} ln x^{2} + 1 + 2 arctan (x)

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

定数を解答に追加する

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

グラフ

人気の例

integral of sec(3x+5)

\int sec (3 x + 5) d x

integral of cos(2+x)

\int cos (2 + x) d x

integral of-xsin(3x^2)

\int - x sin (3 x^{2}) d x

integral of 7(7x+4)^5

\int 7 (7 x + 4)^{5} d x

integral of y/((y^2+x^2)^{3/2)}

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