Without the use of a calculator, show that \[\int_{2-\sqrt{3}}^{5}\frac{\left|x-5\right|}{x^2-4x+7}\

Question

Without the use of a calculator, show that $$\int_{2-\sqrt{3}}^{5}\frac{\left|x-5\right|}{x^2-4x+7}\,\mathrm{d}x=p\ln 2+q\pi,$$ where $p$ and $q$ are exact real constants to be determined.[6]

Accepted Answer

$p=-\frac{1}{2},\quad q=\frac{7\sqrt{3}}{12}$.

Exact Definite Integral with a Modulus

Exact Definite Integral with a Modulus