What is the integral of (6x+5)/(x^2+3x-4) ?

The integral of (6x+5)/(x^2+3x-4) is 3ln|4x^2+12x-16|+4/5 (ln| 8/5+2/5 x|-ln|-2/5+2/5 x|)+C

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integral of (6x+5)/(x^2+3x-4)

\int \frac{6 x + 5}{x ^{2} + 3 x - 4} d x

3 ln 4 x^{2} + 12 x - 16 + \frac{4}{5} (ln \frac{8}{5} + \frac{2}{5} x - ln - \frac{2}{5} + \frac{2}{5} x) + C

Solution steps

\int \frac{6 x + 5}{x ^{2} + 3 x - 4} d x

Complete the square

x^{2} + 3 x - 4 : (x + \frac{3}{2})^{2} - \frac{25}{4}

= \int \frac{6 x + 5}{( x + \frac{3}{2} ) ^{2} - \frac{25}{4}} d x

Apply u-substitution

: \int \frac{4 ( 6 u - 4 )}{4 u ^{2} - 25} d u

= \int \frac{4 ( 6 u - 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{6 u - 4}{4 u ^{2} - 25} d u

Expand

\frac{6 u - 4}{4 u ^{2} - 25} : \frac{6 u}{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{6 u}{4 u ^{2} - 25} d u - \int \frac{4}{4 u ^{2} - 25} d u)

\int \frac{6 u}{4 u ^{2} - 25} d u = \frac{3}{4} ln 4 u^{2} - 25

\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 (\frac{3}{4} ln 4 u^{2} - 25 - (- \frac{1}{5} (ln \frac{2}{5} u + 1 - ln \frac{2}{5} u - 1)))

Substitute back

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

= 4 (\frac{3}{4} ln 4 (x + \frac{3}{2})^{2} - 25 - (- \frac{1}{5} (ln \frac{2}{5} (x + \frac{3}{2}) + 1 - ln \frac{2}{5} (x + \frac{3}{2}) - 1)))

Simplify

4 (\frac{3}{4} ln 4 (x + \frac{3}{2})^{2} - 25 - (- \frac{1}{5} (ln \frac{2}{5} (x + \frac{3}{2}) + 1 - ln \frac{2}{5} (x + \frac{3}{2}) - 1))) : 3 ln 4 x^{2} + 12 x - 16 + \frac{4}{5} (ln \frac{8}{5} + \frac{2}{5} x - ln - \frac{2}{5} + \frac{2}{5} x)

= 3 ln 4 x^{2} + 12 x - 16 + \frac{4}{5} (ln \frac{8}{5} + \frac{2}{5} x - ln - \frac{2}{5} + \frac{2}{5} x)

Add a constant to the solution

= 3 ln 4 x^{2} + 12 x - 16 + \frac{4}{5} (ln \frac{8}{5} + \frac{2}{5} x - ln - \frac{2}{5} + \frac{2}{5} x) + C

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What is the integral of (6x+5)/(x^2+3x-4) ?
The integral of (6x+5)/(x^2+3x-4) is 3ln|4x^2+12x-16|+4/5 (ln| 8/5+2/5 x|-ln|-2/5+2/5 x|)+C