Answer
$\displaystyle 2\ln(\frac{3}{2})≈0.810930$
Work Step by Step
Partial Fraction Decomposition: $\displaystyle \quad\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a}+\frac{B}{x-b}$
$\displaystyle \int_0^1\frac{2}{2x^2+3x+1}dx=\int_0^1\frac{2}{(2x+1)(x+1)}dx$
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Partial Fraction Decomposition:
$2=A(2x+1)+B(x+1)$
Let $\displaystyle x=-\frac{1}{2}\qquad \qquad$ Let $x = -1$
$2=B(\displaystyle \frac{1}{2})\qquad\qquad \quad2=A(-1)$
$4=B\qquad\qquad\qquad$ $A=-2$
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$\displaystyle \int_0^1\frac{2}{(2x+1)(x+1)}dx = \int^1_0\frac{4}{2x+1}-\frac{2}{x+1}dx=-\int_0^1\frac{2}{x+1}dx+\int_0^1\frac{4}{2x+1}dx$
$\displaystyle [-2\ln|x+1|]_0^1+\int_0^1\frac{4}{2x+1}dx\qquad$ Let $u=2x+1\quad \rightarrow \quad du=2$ $dx$
$\displaystyle [-2\ln|x+1|]_0^1+4\cdot\frac{1}{2}\int_1^3\frac{du}{u}$
$\displaystyle [-2\ln|x+1|+2\ln|2x+1|]_0^1 = 2[\ln(2x+1)-\ln(x+1)]_0^1 = 2[\ln(\frac{2x+1}{x+1})]_0^1$
$\displaystyle 2([\ln(\frac{2(1)+1}{1+1})]-[\ln(\frac{2(0)+1}{0+1})])$
$\displaystyle 2[\ln(\frac{3}{2})-\ln(\frac{1}{1})]$
$\displaystyle 2\ln(\frac{3}{2})≈0.810930$
