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# Integrate the function$\frac{5x}{(x+1)(x^2+9)}$

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Toolbox:
• If the function is of a rational form $\frac{1}{(x+a)(x^2+b)}$ then it can be resolved into its partial of the form $\frac{A}{x+a}+\frac{Bx+C}{x^2+b}$
• (ii)$\int\frac{dx}{x+a}=log|x+a|+c$
Given:$I=\frac{5x}{(x+1)(x^2+9)}$

The given rational function is a proper rational function, and it can be resolved as

$\large\frac{5x}{(x+a)(x^2+9)}=\frac{A}{(x+1)}+\frac{Bx+c}{x^2+9}$

$=>5x=A((x^2+9)+(Bx+c)(x+1)$

Equating the coefficient of $x^2$.

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