Question

Evaluate:- $\int \frac{1}{3x^2+5x+7}dx.$

Answer 1

$\int \frac{dx}{3x^2+5x+7}$

$\frac{1}{3}\int \frac{dx}{x^2+\frac{5}{3}x+\frac{7}{3}}$

$\frac{1}{3}\int \frac{dx}{{\left(x+\frac{5}{6}\right)}^2+{\left(\frac{\sqrt{59}}{6}\right)}^2}\because \int \frac{dx}{x^2+a^2}=\frac{{\tan }^{-1}}{a}\frac{x}{a}$

$\frac{1}{3}\times \frac{1}{\left(\sqrt{59}/6\right)}{\tan }^{-1}\frac{\left(x+5/6\right)}{\sqrt{59}/6}+c$

$\frac{2}{\sqrt{59}}{\tan }^{-1}\frac{\left(6x+5\right)}{\sqrt{59}}+c$