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人気のある微分積分 >

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

解

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

解

2 (\frac{1}{4} ln 4 x^{2} + 20 x - 24 + \frac{1}{28} (ln \frac{12}{7} + \frac{2}{7} x - ln - \frac{2}{7} + \frac{2}{7} x)) + C

解答ステップ

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

平方完成する

x^{2} + 5 x - 6 : (x + \frac{5}{2})^{2} - \frac{49}{4}

= \int \frac{x + 2}{( x + \frac{5}{2} ) ^{2} - \frac{49}{4}} d x

u置換積分法を適用する

= \int \frac{2 ( 2 u - 1 )}{4 u ^{2} - 49} 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 - 1}{4 u ^{2} - 49} d u

拡張

\frac{2 u - 1}{4 u ^{2} - 49} : \frac{2 u}{4 u ^{2} - 49} - \frac{1}{4 u ^{2} - 49}

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} - 49} d u - \int \frac{1}{4 u ^{2} - 49} d u)

\int \frac{2 u}{4 u ^{2} - 49} d u = \frac{1}{4} ln 4 u^{2} - 49

\int \frac{1}{4 u ^{2} - 49} d u = - \frac{1}{28} (ln \frac{2}{7} u + 1 - ln \frac{2}{7} u - 1)

= 2 (\frac{1}{4} ln 4 u^{2} - 49 - (- \frac{1}{28} (ln \frac{2}{7} u + 1 - ln \frac{2}{7} u - 1)))

代用を戻す

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

= 2 (\frac{1}{4} ln 4 (x + \frac{5}{2})^{2} - 49 - (- \frac{1}{28} (ln \frac{2}{7} (x + \frac{5}{2}) + 1 - ln \frac{2}{7} (x + \frac{5}{2}) - 1)))

簡素化

2 (\frac{1}{4} ln 4 (x + \frac{5}{2})^{2} - 49 - (- \frac{1}{28} (ln \frac{2}{7} (x + \frac{5}{2}) + 1 - ln \frac{2}{7} (x + \frac{5}{2}) - 1))) : 2 (\frac{1}{4} ln 4 x^{2} + 20 x - 24 + \frac{1}{28} (ln \frac{12}{7} + \frac{2}{7} x - ln - \frac{2}{7} + \frac{2}{7} x))

= 2 (\frac{1}{4} ln 4 x^{2} + 20 x - 24 + \frac{1}{28} (ln \frac{12}{7} + \frac{2}{7} x - ln - \frac{2}{7} + \frac{2}{7} x))

定数を解答に追加する

= 2 (\frac{1}{4} ln 4 x^{2} + 20 x - 24 + \frac{1}{28} (ln \frac{12}{7} + \frac{2}{7} x - ln - \frac{2}{7} + \frac{2}{7} x)) + C

グラフ

人気の例

integral from 0 to 1 of x^2x

\int_{0}^{1} x^{2} x d x

derivative x/(x^2-7)

d er i v a t i v e \frac{x}{x ^{2} - 7}

integral of (x^4-x^2+1/(x^3)-1/(x^2))

\int (x^{4} - x^{2} + \frac{1}{x ^{3}} - \frac{1}{x ^{2}}) d x

integral of 3+2xy

\int 3 + 2 x y d x

derivative y=-2x-4

d er i v a t i v e y = - 2 x - 4