First slide

Methods of integration

Question

$Evaluate \int \frac{4 x + 1}{x^{2} + 3 x + 2} d x$

Moderate

A

$2 \log |x^{2} + 3 x + 2| - \frac{5}{2} \log |\frac{x + 1}{x + 2}| + C$
B

$2 \log |x^{2} + 3 x + 2| - 5 \log |\frac{2 x + 1}{2 x + 2}| + C$
C

$2 \log |x^{2} + 3 x + 2| - 5 \log |\frac{x + 1}{x + 2}| + C$
D

None of these

Solution

$\begin{array}{l} I = \int \frac{4 x + 1}{x^{2} + 3 x + 2} d x \\ = \int \frac{2 (2 x + 3) - 5}{x^{2} + 3 x + 2} d x \\ = 2 \int \frac{2 x + 3}{x^{2} + 3 x + 2} d x - 5 \int \frac{1}{x^{2} + 3 x + 2} d x \\ = 2 \log x^{2} + 3 x + 2 ∣ - 5 \int \frac{1}{x^{2} + 3 x + (\frac{9}{4}) - (\frac{9}{4}) + 2} d x \\ = 2 \log |x^{2} + 3 x + 2| - 5 \int \frac{1}{(x + \frac{3}{2})^{2} - (\frac{1}{2})^{2}} d x \\ \begin{array}{l} = 2 \log |x^{2} + 3 x + 2| - 5 \frac{1}{2 (\frac{1}{2})} \log |\frac{x + \frac{3}{2} - \frac{1}{2}}{x + \frac{3}{2} + \frac{1}{2}}| + C \\ = 2 \log |x^{2} + 3 x + 2| - 5 \log |\frac{x + 1}{x + 2}| + C \end{array} \end{array}$

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