Question:

<p>Express $3x + 2 = \ell(\text{d.c. of } 4x^2 + 4x + 5) + m$</p><p>or, $3x + 2 = \ell(8x + 4) + m$</p><p>Comparing the coefficients, we get</p><p>$8\ell = 3$ and $4\ell + m = 2 \Rightarrow \ell = 3/8$ and $m = 2 - 4\ell = 1/2$</p><p>$\Rightarrow I = \int \frac{3x+2}{4x^2+4x+5} dx = \frac{3}{8} \int \frac{8x+4}{4x^2+4x+5} dx + \frac{1}{2} \int \frac{dx}{4x^2+4x+5}$</p><p>$= \frac{3}{8} \log |4x^2 + 4x + 5| + \frac{1}{8} \int \frac{dx}{x^2 + x + \frac{5}{4}}$</p><p>$= \frac{3}{8} \log |4x^2 + 4x + 5| + \frac{1}{8} \tan^{-1} \left( x + \frac{1}{2} \right) + C$</p>