What is the integral of (2x)/(x^2-3x+2) ?

The integral of (2x)/(x^2-3x+2) is ln|4x^2-12x+8|-3ln|2x-2|+3ln|2x-4|+C

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

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

ln 4 x^{2} - 12 x + 8 - 3 ln ∣ 2 x - 2 ∣ + 3 ln ∣ 2 x - 4 ∣ + C

Solution steps

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

Take the constant out:

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

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

Complete the square

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{2 ( 2 u + 3 )}{4 u ^{2} - 1} d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int \frac{2 u + 3}{4 u ^{2} - 1} d u

Expand

\frac{2 u + 3}{4 u ^{2} - 1} : \frac{2 u}{4 u ^{2} - 1} + \frac{3}{4 u ^{2} - 1}

Apply the Sum Rule:

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

= 2 \cdot 2 (\int \frac{2 u}{4 u ^{2} - 1} d u + \int \frac{3}{4 u ^{2} - 1} d u)

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

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

= 2 \cdot 2 (\frac{1}{4} ln 4 u^{2} - 1 - \frac{3}{4} (ln ∣ 2 u + 1 ∣ - ln ∣ 2 u - 1 ∣))

Substitute back

u = x - \frac{3}{2}

= 2 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{3}{2})^{2} - 1 - \frac{3}{4} (ln 2 (x - \frac{3}{2}) + 1 - ln 2 (x - \frac{3}{2}) - 1))

Simplify

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

= ln 4 x^{2} - 12 x + 8 - 3 ln ∣ 2 x - 2 ∣ + 3 ln ∣ 2 x - 4 ∣

Add a constant to the solution

= ln 4 x^{2} - 12 x + 8 - 3 ln ∣ 2 x - 2 ∣ + 3 ln ∣ 2 x - 4 ∣ + C

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What is the integral of (2x)/(x^2-3x+2) ?
The integral of (2x)/(x^2-3x+2) is ln|4x^2-12x+8|-3ln|2x-2|+3ln|2x-4|+C