What is the integral of (5x-4)/(x^2+7x+6) ?

The integral of (5x-4)/(x^2+7x+6) is

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integral of (5x-4)/(x^2+7x+6)

\int \frac{5 x - 4}{x ^{2} + 7 x + 6} d x

4 (5 (\frac{1}{8} ln 4 x^{2} + 28 x + 24 + \frac{7}{40} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x)) + \frac{1}{5} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x)) + C

Solution steps

\int \frac{5 x - 4}{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{5 x - 4}{( x + \frac{7}{2} ) ^{2} - \frac{25}{4}} d x

Apply u-substitution

: \int \frac{4 ( 5 ( u - \frac{7}{2} ) - 4 )}{4 u ^{2} - 25} d u

= \int \frac{4 ( 5 ( u - \frac{7}{2} ) - 4 )}{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{5 ( u - \frac{7}{2} ) - 4}{4 u ^{2} - 25} d u

Expand

\frac{5 ( u - \frac{7}{2} ) - 4}{4 u ^{2} - 25} : \frac{5 ( u - \frac{7}{2} )}{4 u ^{2} - 25} - \frac{4}{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{5 ( u - \frac{7}{2} )}{4 u ^{2} - 25} d u - \int \frac{4}{4 u ^{2} - 25} d u)

\int \frac{5 ( u - \frac{7}{2} )}{4 u ^{2} - 25} d u = 5 (\frac{1}{8} ln 4 u^{2} - 25 + \frac{7}{40} (ln \frac{2}{5} u + 1 - ln \frac{2}{5} u - 1))

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

= 4 (5 (\frac{1}{8} ln 4 u^{2} - 25 + \frac{7}{40} (ln \frac{2}{5} u + 1 - ln \frac{2}{5} u - 1)) - (- \frac{1}{5} (ln \frac{2}{5} u + 1 - ln \frac{2}{5} u - 1)))

Substitute back

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

= 4 (5 (\frac{1}{8} ln 4 (x + \frac{7}{2})^{2} - 25 + \frac{7}{40} (ln \frac{2}{5} (x + \frac{7}{2}) + 1 - ln \frac{2}{5} (x + \frac{7}{2}) - 1)) - (- \frac{1}{5} (ln \frac{2}{5} (x + \frac{7}{2}) + 1 - ln \frac{2}{5} (x + \frac{7}{2}) - 1)))

Simplify

4 (5 (\frac{1}{8} ln 4 (x + \frac{7}{2})^{2} - 25 + \frac{7}{40} (ln \frac{2}{5} (x + \frac{7}{2}) + 1 - ln \frac{2}{5} (x + \frac{7}{2}) - 1)) - (- \frac{1}{5} (ln \frac{2}{5} (x + \frac{7}{2}) + 1 - ln \frac{2}{5} (x + \frac{7}{2}) - 1))) : 4 (5 (\frac{1}{8} ln 4 x^{2} + 28 x + 24 + \frac{7}{40} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x)) + \frac{1}{5} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x))

= 4 (5 (\frac{1}{8} ln 4 x^{2} + 28 x + 24 + \frac{7}{40} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x)) + \frac{1}{5} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x))

Add a constant to the solution

= 4 (5 (\frac{1}{8} ln 4 x^{2} + 28 x + 24 + \frac{7}{40} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x)) + \frac{1}{5} (ln \frac{12}{5} + \frac{2}{5} x - ln \frac{2}{5} + \frac{2}{5} x)) + C

x^{\frac{3}{2}}

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What is the integral of (5x-4)/(x^2+7x+6) ?
The integral of (5x-4)/(x^2+7x+6) is