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Functions

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Question

Solve : $\int \frac{x^{2} + x + 5}{3 x + 2} d x$

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Solution

$\frac{x^{2} + x + 5}{3 x + 2}$

$= \frac{x}{3} + \frac{\frac{x}{3} + 5}{3 x + 2}$

$= \frac{x}{3} + \frac{1}{9} + \frac{\frac{43}{9}}{3 x + 2}$

$\frac{x^{2} + x + 5}{3 x + 2} = \frac{x}{3} + \frac{1}{9} + \frac{43}{9 (3 x + 2)}$

$\int \frac{x^{2} + x + 5}{3 x + 2} d x = \int (\frac{x}{3} + \frac{1}{9} + \frac{43}{9 (3 x + 2)}) d x$

$= \frac{x^{2}}{6} + \frac{x}{9} + \frac{43}{9} (\frac{1}{3} ln | 3 x + 2 |) + C$

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