What is the integral of (x+12)/(3x^2+7x+2) ?

The integral of (x+12)/(3x^2+7x+2) is

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integral of (x+12)/(3x^2+7x+2)

\int \frac{x + 12}{3 x ^{2} + 7 x + 2} d x

2 (\frac{1}{12} ln 36 x^{2} + 84 x + 24 + \frac{7}{60} (ln \frac{12}{5} + \frac{6}{5} x - ln \frac{2}{5} + \frac{6}{5} x)) - \frac{12}{5} (ln \frac{6}{5} x + \frac{12}{5} - ln \frac{6}{5} x + \frac{2}{5}) + C

Solution steps

\int \frac{x + 12}{3 x ^{2} + 7 x + 2} d x

Expand

\frac{x + 12}{3 x ^{2} + 7 x + 2} : \frac{x}{3 x ^{2} + 7 x + 2} + \frac{12}{3 x ^{2} + 7 x + 2}

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{x}{3 x ^{2} + 7 x + 2} d x + \int \frac{12}{3 x ^{2} + 7 x + 2} d x

\int \frac{x}{3 x ^{2} + 7 x + 2} d x = 2 (\frac{1}{12} ln 36 x^{2} + 84 x + 24 + \frac{7}{60} (ln \frac{12}{5} + \frac{6}{5} x - ln \frac{2}{5} + \frac{6}{5} x))

\int \frac{12}{3 x ^{2} + 7 x + 2} d x = - \frac{12}{5} (ln \frac{6}{5} x + \frac{12}{5} - ln \frac{6}{5} x + \frac{2}{5})

= 2 (\frac{1}{12} ln 36 x^{2} + 84 x + 24 + \frac{7}{60} (ln \frac{12}{5} + \frac{6}{5} x - ln \frac{2}{5} + \frac{6}{5} x)) - \frac{12}{5} (ln \frac{6}{5} x + \frac{12}{5} - ln \frac{6}{5} x + \frac{2}{5})

Add a constant to the solution

= 2 (\frac{1}{12} ln 36 x^{2} + 84 x + 24 + \frac{7}{60} (ln \frac{12}{5} + \frac{6}{5} x - ln \frac{2}{5} + \frac{6}{5} x)) - \frac{12}{5} (ln \frac{6}{5} x + \frac{12}{5} - ln \frac{6}{5} x + \frac{2}{5}) + C

x \to 2 lim (\frac{250}{0})

\int 4 x^{2} d x

n = 1 \sum \infty \frac{1}{2 n ( n + 1 )}

\frac{d}{d x} (\frac{sin ( 5 x )}{e ^{3 x}})

y = x^{2}, y = 6 - x

What is the integral of (x+12)/(3x^2+7x+2) ?
The integral of (x+12)/(3x^2+7x+2) is