What is the integral of (2x-3)/(x^2+5x+4) ?

The integral of (2x-3)/(x^2+5x+4) is ln|4x^2+20x+16|+8/3 (ln| 8/3+2/3 x|-ln| 2/3+2/3 x|)+C

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integral of (2x-3)/(x^2+5x+4)

\int \frac{2 x - 3}{x ^{2} + 5 x + 4} d x

ln 4 x^{2} + 20 x + 16 + \frac{8}{3} (ln \frac{8}{3} + \frac{2}{3} x - ln \frac{2}{3} + \frac{2}{3} x) + C

Solution steps

\int \frac{2 x - 3}{x ^{2} + 5 x + 4} d x

Complete the square

x^{2} + 5 x + 4 : (x + \frac{5}{2})^{2} - \frac{9}{4}

= \int \frac{2 x - 3}{( x + \frac{5}{2} ) ^{2} - \frac{9}{4}} d x

Apply u-substitution

: \int \frac{4 ( 2 u - 8 )}{4 u ^{2} - 9} d u

= \int \frac{4 ( 2 u - 8 )}{4 u ^{2} - 9} 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 - 8}{4 u ^{2} - 9} d u

Expand

\frac{2 u - 8}{4 u ^{2} - 9} : \frac{2 u}{4 u ^{2} - 9} - \frac{8}{4 u ^{2} - 9}

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} - 9} d u - \int \frac{8}{4 u ^{2} - 9} d u)

\int \frac{2 u}{4 u ^{2} - 9} d u = \frac{1}{4} ln 4 u^{2} - 9

\int \frac{8}{4 u ^{2} - 9} d u = - \frac{2}{3} (ln \frac{2}{3} u + 1 - ln \frac{2}{3} u - 1)

= 4 (\frac{1}{4} ln 4 u^{2} - 9 - (- \frac{2}{3} (ln \frac{2}{3} u + 1 - ln \frac{2}{3} u - 1)))

Substitute back

u = x + \frac{5}{2}

= 4 (\frac{1}{4} ln 4 (x + \frac{5}{2})^{2} - 9 - (- \frac{2}{3} (ln \frac{2}{3} (x + \frac{5}{2}) + 1 - ln \frac{2}{3} (x + \frac{5}{2}) - 1)))

Simplify

4 (\frac{1}{4} ln 4 (x + \frac{5}{2})^{2} - 9 - (- \frac{2}{3} (ln \frac{2}{3} (x + \frac{5}{2}) + 1 - ln \frac{2}{3} (x + \frac{5}{2}) - 1))) : ln 4 x^{2} + 20 x + 16 + \frac{8}{3} (ln \frac{8}{3} + \frac{2}{3} x - ln \frac{2}{3} + \frac{2}{3} x)

= ln 4 x^{2} + 20 x + 16 + \frac{8}{3} (ln \frac{8}{3} + \frac{2}{3} x - ln \frac{2}{3} + \frac{2}{3} x)

Add a constant to the solution

= ln 4 x^{2} + 20 x + 16 + \frac{8}{3} (ln \frac{8}{3} + \frac{2}{3} x - ln \frac{2}{3} + \frac{2}{3} x) + C

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What is the integral of (2x-3)/(x^2+5x+4) ?
The integral of (2x-3)/(x^2+5x+4) is ln|4x^2+20x+16|+8/3 (ln| 8/3+2/3 x|-ln| 2/3+2/3 x|)+C