What is the integral of (2x)/(x^2-7x+12) ?

The integral of (2x)/(x^2-7x+12) is ln|4x^2-28x+48|-7ln|2x-6|+7ln|2x-8|+C

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integral of (2x)/(x^2-7x+12)

\int \frac{2 x}{x ^{2} - 7 x + 12} d x

ln 4 x^{2} - 28 x + 48 - 7 ln ∣ 2 x - 6 ∣ + 7 ln ∣ 2 x - 8 ∣ + C

Solution steps

\int \frac{2 x}{x ^{2} - 7 x + 12} 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} - 7 x + 12} d x

Complete the square

x^{2} - 7 x + 12 : (x - \frac{7}{2})^{2} - \frac{1}{4}

= 2 \cdot \int \frac{x}{( x - \frac{7}{2} ) ^{2} - \frac{1}{4}} d x

Apply u-substitution

: \int \frac{2 ( 2 u + 7 )}{4 u ^{2} - 1} d u

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

Expand

\frac{2 u + 7}{4 u ^{2} - 1} : \frac{2 u}{4 u ^{2} - 1} + \frac{7}{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{7}{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{7}{4 u ^{2} - 1} d u = - \frac{7}{4} (ln ∣ 2 u + 1 ∣ - ln ∣ 2 u - 1 ∣)

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

Substitute back

u = x - \frac{7}{2}

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

Simplify

2 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{7}{2})^{2} - 1 - \frac{7}{4} (ln 2 (x - \frac{7}{2}) + 1 - ln 2 (x - \frac{7}{2}) - 1)) : ln 4 x^{2} - 28 x + 48 - 7 ln ∣ 2 x - 6 ∣ + 7 ln ∣ 2 x - 8 ∣

= ln 4 x^{2} - 28 x + 48 - 7 ln ∣ 2 x - 6 ∣ + 7 ln ∣ 2 x - 8 ∣

Add a constant to the solution

= ln 4 x^{2} - 28 x + 48 - 7 ln ∣ 2 x - 6 ∣ + 7 ln ∣ 2 x - 8 ∣ + C

\frac{d y}{d x} = \frac{2 + y}{x + 1}

f (x) = 5 x^{2} - x^{3}, (1, 4)

f^{^{'}} (x) = tan (x)

\int \frac{4 x ^{2} - 19 x - 27}{( x + 1 ) ^{2} ( x - 3 )} d x

lo g_{4} (x)

What is the integral of (2x)/(x^2-7x+12) ?
The integral of (2x)/(x^2-7x+12) is ln|4x^2-28x+48|-7ln|2x-6|+7ln|2x-8|+C