<p>Evaluate \(\displaystyle\int_0^1\frac{dx}{x^2+3x+2}\)</p>
Step-by-Step Solution
Key Concept: Factor: x^2+3x+2 = (x+1)(x+2). Partial fractions: 1/[(x+1)(x+2)] = 1/(x+1) - 1/(x+2).
<div class='solution'>
<p>$\dfrac{1}{(x+1)(x+2)} = \dfrac{1}{x+1}-\dfrac{1}{x+2}$</p>
<p>$$I = \int_0^1\left(\frac{1}{x+1}-\frac{1}{x+2}\right)dx = [\ln(x+1)-\ln(x+2)]_0^1$$</p>
<p>$$= (\ln 2 - \ln 3)-(\ln 1 - \ln 2) = \ln 2 - \ln 3 + \ln 2 = 2\ln 2 - \ln 3 = \ln\frac{4}{3}$$</p>
<p>$$\boxed{I = \ln\frac{4}{3}}$$</p>
Correct Answer: A