What is the integral of (3x+2)/(sqrt(19-5x+x^2)) ?

The integral of (3x+2)/(sqrt(19-5x+x^2)) is

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integral of (3x+2)/(sqrt(19-5x+x^2))

\int \frac{3 x + 2}{19 - 5 x + x ^{2}} d x

3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5) + C

Solution steps

\int \frac{3 x + 2}{19 - 5 x + x ^{2}} d x

Expand

\frac{3 x + 2}{19 - 5 x + x ^{2}} : \frac{3 x}{19 - 5 x + x ^{2}} + \frac{2}{19 - 5 x + x ^{2}}

= \int \frac{3 x}{19 - 5 x + x ^{2}} + \frac{2}{19 - 5 x + x ^{2}} d x

Apply the Sum Rule:

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

= \int \frac{3 x}{19 - 5 x + x ^{2}} d x + \int \frac{2}{19 - 5 x + x ^{2}} d x

\int \frac{3 x}{19 - 5 x + x ^{2}} d x = 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{4 x ^{2} - 20 x + 76}{51})

\int \frac{2}{19 - 5 x + x ^{2}} d x = 2 ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)

= 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{4 x ^{2} - 20 x + 76}{51}) + 2 ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)

Simplify

3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{4 x ^{2} - 20 x + 76}{51}) + 2 ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76) : 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5)

= 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5)

Add a constant to the solution

= 3 (x^{2} - 5 x + 19 + \frac{5}{2} ln \frac{2}{51} x - \frac{5}{51} + \frac{1}{51} (4 x^{2} - 20 x + 76)) + 2 ln \frac{1}{51} (4 x^{2} - 20 x + 76) + \frac{1}{51} (2 x - 5) + C

What is the integral of (3x+2)/(sqrt(19-5x+x^2)) ?
The integral of (3x+2)/(sqrt(19-5x+x^2)) is