What is the integral of (2x-3)/(x^2+7x+6) ?

The integral of (2x-3)/(x^2+7x+6) is ln|4x^2+28x+24|+2ln| 2/5 x+12/5 |-2ln| 2/5 x+2/5 |+C

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integral of (2x-3)/(x^2+7x+6)

\int \frac{2 x - 3}{x ^{2} + 7 x + 6} d x

ln 4 x^{2} + 28 x + 24 + 2 ln \frac{2}{5} x + \frac{12}{5} - 2 ln \frac{2}{5} x + \frac{2}{5} + C

Solution steps

\int \frac{2 x - 3}{x ^{2} + 7 x + 6} d x

Complete the square

x^{2} + 7 x + 6 : (x + \frac{7}{2})^{2} - \frac{25}{4}

= \int \frac{2 x - 3}{( x + \frac{7}{2} ) ^{2} - \frac{25}{4}} d x

Apply u-substitution

: \int \frac{4 ( 2 u - 10 )}{4 u ^{2} - 25} d u

= \int \frac{4 ( 2 u - 10 )}{4 u ^{2} - 25} d u

Take the constant out:

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

= 4 \cdot \int \frac{2 u - 10}{4 u ^{2} - 25} d u

Expand

\frac{2 u - 10}{4 u ^{2} - 25} : \frac{2 u}{4 u ^{2} - 25} - \frac{10}{4 u ^{2} - 25}

Apply the Sum Rule:

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

= 4 (\int \frac{2 u}{4 u ^{2} - 25} d u - \int \frac{10}{4 u ^{2} - 25} d u)

\int \frac{2 u}{4 u ^{2} - 25} d u = \frac{1}{4} ln 4 u^{2} - 25

\int \frac{10}{4 u ^{2} - 25} d u = - \frac{1}{2} ln \frac{2}{5} u + 1 + \frac{1}{2} ln \frac{2}{5} u - 1

= 4 (\frac{1}{4} ln 4 u^{2} - 25 - (- \frac{1}{2} ln \frac{2}{5} u + 1 + \frac{1}{2} ln \frac{2}{5} u - 1))

Substitute back

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

= 4 (\frac{1}{4} ln 4 (x + \frac{7}{2})^{2} - 25 - (- \frac{1}{2} ln \frac{2}{5} (x + \frac{7}{2}) + 1 + \frac{1}{2} ln \frac{2}{5} (x + \frac{7}{2}) - 1))

Simplify

4 (\frac{1}{4} ln 4 (x + \frac{7}{2})^{2} - 25 - (- \frac{1}{2} ln \frac{2}{5} (x + \frac{7}{2}) + 1 + \frac{1}{2} ln \frac{2}{5} (x + \frac{7}{2}) - 1)) : ln 4 x^{2} + 28 x + 24 + 2 ln \frac{2}{5} x + \frac{12}{5} - 2 ln \frac{2}{5} x + \frac{2}{5}

= ln 4 x^{2} + 28 x + 24 + 2 ln \frac{2}{5} x + \frac{12}{5} - 2 ln \frac{2}{5} x + \frac{2}{5}

Add a constant to the solution

= ln 4 x^{2} + 28 x + 24 + 2 ln \frac{2}{5} x + \frac{12}{5} - 2 ln \frac{2}{5} x + \frac{2}{5} + C

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What is the integral of (2x-3)/(x^2+7x+6) ?
The integral of (2x-3)/(x^2+7x+6) is ln|4x^2+28x+24|+2ln| 2/5 x+12/5 |-2ln| 2/5 x+2/5 |+C