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

解答

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

x + \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{15}{2} ln ∣ 2 x - 2 ∣ + \frac{15}{2} ln ∣ 2 x - 4 ∣ - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + C

求解步骤

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

乘开

\frac{x ^{2} + 3 x + 3}{x ^{2} - 3 x + 2} : \frac{x ^{2}}{x ^{2} - 3 x + 2} + \frac{3 x}{x ^{2} - 3 x + 2} + \frac{3}{x ^{2} - 3 x + 2}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{x ^{2}}{x ^{2} - 3 x + 2} d x = x - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣

\int \frac{3 x}{x ^{2} - 3 x + 2} d x = 6 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣))

\int \frac{3}{x ^{2} - 3 x + 2} d x = - 6 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣)

= x - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + 6 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣)) - 6 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣)

化简

x - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + 6 (\frac{1}{4} ln 4 x^{2} - 12 x + 8 - \frac{3}{4} (ln ∣ 2 x - 2 ∣ - ln ∣ 2 x - 4 ∣)) - 6 (\frac{1}{2} ln ∣ 2 x - 2 ∣ - \frac{1}{2} ln ∣ 2 x - 4 ∣) : x + \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{15}{2} ln ∣ 2 x - 2 ∣ + \frac{15}{2} ln ∣ 2 x - 4 ∣ - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣

= x + \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{15}{2} ln ∣ 2 x - 2 ∣ + \frac{15}{2} ln ∣ 2 x - 4 ∣ - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣

解答补常数

= x + \frac{3}{2} ln 4 x^{2} - 12 x + 8 - \frac{15}{2} ln ∣ 2 x - 2 ∣ + \frac{15}{2} ln ∣ 2 x - 4 ∣ - ln ∣ x - 1 ∣ + 4 ln ∣ x - 2 ∣ + C

\int (2 x + 5) x + 3 d x

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\int y e^{(t - 1) y} d y

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