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人気のある微分積分 >

integral of x/(x^2-5x+3)

解

\int \frac{x}{x ^{2} - 5 x + 3} d x

解

2 (\frac{1}{4} ln 4 x^{2} - 20 x + 12 - \frac{5}{4 13} (ln \frac{2 x - 5}{13} + 1 - ln \frac{2 x - 5}{13} - 1)) + C

解答ステップ

\int \frac{x}{x ^{2} - 5 x + 3} d x

平方完成する

x^{2} - 5 x + 3 : (x - \frac{5}{2})^{2} - \frac{13}{4}

= \int \frac{x}{( x - \frac{5}{2} ) ^{2} - \frac{13}{4}} d x

u置換積分法を適用する

= \int \frac{2 ( 2 u + 5 )}{4 u ^{2} - 13} 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{2 u + 5}{4 u ^{2} - 13} d u

拡張

\frac{2 u + 5}{4 u ^{2} - 13} : \frac{2 u}{4 u ^{2} - 13} + \frac{5}{4 u ^{2} - 13}

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 u}{4 u ^{2} - 13} d u + \int \frac{5}{4 u ^{2} - 13} d u)

\int \frac{2 u}{4 u ^{2} - 13} d u = \frac{1}{4} ln 4 u^{2} - 13

\int \frac{5}{4 u ^{2} - 13} d u = - \frac{5}{4 13} (ln \frac{2}{13} u + 1 - ln \frac{2}{13} u - 1)

= 2 (\frac{1}{4} ln 4 u^{2} - 13 - \frac{5}{4 13} (ln \frac{2}{13} u + 1 - ln \frac{2}{13} u - 1))

代用を戻す

u = x - \frac{5}{2}

= 2 (\frac{1}{4} ln 4 (x - \frac{5}{2})^{2} - 13 - \frac{5}{4 13} (ln \frac{2}{13} (x - \frac{5}{2}) + 1 - ln \frac{2}{13} (x - \frac{5}{2}) - 1))

簡素化

2 (\frac{1}{4} ln 4 (x - \frac{5}{2})^{2} - 13 - \frac{5}{4 13} (ln \frac{2}{13} (x - \frac{5}{2}) + 1 - ln \frac{2}{13} (x - \frac{5}{2}) - 1)) : 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 12 - \frac{5}{4 13} (ln \frac{2 x - 5}{13} + 1 - ln \frac{2 x - 5}{13} - 1))

= 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 12 - \frac{5}{4 13} (ln \frac{2 x - 5}{13} + 1 - ln \frac{2 x - 5}{13} - 1))

定数を解答に追加する

= 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 12 - \frac{5}{4 13} (ln \frac{2 x - 5}{13} + 1 - ln \frac{2 x - 5}{13} - 1)) + C

グラフ

人気の例

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

\int \frac{2 x ^{2}}{x ^{2} + 3} d x

integral of (2x+7)(x^2+7x+1)

\int (2 x + 7) (x^{2} + 7 x + 1) d x

integral of ((6x-5))/(sqrt(3x^2-5x+6))

\int \frac{( 6 x - 5 )}{3 x ^{2} - 5 x + 6} d x

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

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

integral of sqrt(3^2-x^2)

\int 3^{2} - x^{2} d x