What is \(\displaystyle\int_0^1 \dfrac{dx}{(ax+bx+c)^2}\) equal to?
\(\dfrac{1}{c(a+b+c)}\)
Writing \(ax+bx+c=(a+b)x+c\) and using \(\displaystyle\int\dfrac{dx}{\{(a+b)x+c\}^2}=-\dfrac{1}{(a+b)\{(a+b)x+c\}}\), evaluating between \(0\) and \(1\) gives \(-\dfrac{1}{(a+b)(a+b+c)}+\dfrac{1}{(a+b)c}=\dfrac{1}{c(a+b+c)}\).
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