Find the integral: d x x^ 2 -6 x+13 - Vidyadip

Question

Find the integral: $\int \frac{d x}{x^{2}-6 x+13}$

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Answer

We have $x^2 - 6x + 13 = x^2 - 6x + 3^2 - 3^2 + 13 = (x - 3)^2 + 4$
So, $\int \frac{d x}{x^{2}-6 x+13}=\int \frac{1}{(x-3)^{2}+2^{2}} d x$
Let x – 3 = t $\Rightarrow$ dx = dt
Therefore,
$\int \frac{d x}{x^{2}-6 x+13}=\int \frac{d t}{t^{2}+2^{2}}=\frac{1}{2} \tan ^{-1} \frac{t}{2}+\mathrm{C}$
$=\frac{1}{2} \tan ^{-1} \frac{x-3}{2}+C$

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