202202071146 Solution to 1969-CE-AMATH-I-XX

Find $\displaystyle{\int\frac{2x+5}{(x^2+5x+1)^3}\,\mathrm{d}x}$ .

Solution.

Set-up.

$\begin{aligned} \textrm{Let }u & =x^2+5x+1 \\ \frac{\mathrm{d}u}{\mathrm{d}x} & = 2x+5 \\ \mathrm{d}x & = \frac{\mathrm{d}u}{2x+5} \\ \end{aligned}$

Then,

$\begin{aligned} &\quad \int\frac{2x+5}{(x^2+5x+1)^3}\,\mathrm{d}x \\ & = \int \frac{2x+5}{u^3}\bigg( \frac{\mathrm{d}u}{2x+5}\bigg) \\ & = \int \frac{1}{u^3} \,\mathrm{d}u \\ & = -\frac{1}{2u^2}+C\textrm{ for some constant }C \\ & = -\frac{1}{2(x^2+5x+1)^2}+C\\ \end{aligned}$

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