Question

(i) Find the integral: $\int \frac{dx}{x^2-6x+13}$

(ii) Find the integral: $\int \frac{dx}{3x^2+13x-10}$

(iii) Find the integral: $\int \frac{dx}{\sqrt{5x^2-2x}}$

Answer 1

(i) $\int \frac{dx}{x^2-6x+13}$

$=\int \frac{dx}{x^2-6x+9+4}$

$=\int \frac{dx}{x^2-\mathrm{2.3.}x+3^2+4}$

$=\int \frac{dx}{(x-3{)}^2+4}=\int \frac{dx}{(x-3{)}^2+2^2}$

Using formula $\frac{dx}{x^2+a^2}=\frac{1}{a}{\tan }^{-1}\frac{x}{a}+C$

$=\frac{1}{2}{\tan }^{-1}\left(\frac{x-3}{2}\right)+C$

Where $C$ is constant of integration.

(ii)$\int \frac{dx}{3x^2+13x-10}$

$\int \frac{dx}{3(x^2+\frac{13}{3}x-\frac{10}{3})}$

$=\frac{1}{3}\int \frac{dx}{x^2+\frac{13}{3}x-\frac{10}{3}}$

$=\frac{1}{3}\int \frac{dx}{x^2+\frac{13}{3}x+{\left(\frac{13}{6}\right)}^2-{\left(\frac{13}{6}\right)}^2-\frac{10}{3}}$

$=\frac{1}{3}\int \frac{dx}{{(x+\frac{13}{6})}^2-\frac{169+120}{36}}$

$=\frac{1}{3}\int \frac{dx}{{(x+\frac{13}{6})}^2-\frac{289}{36}}$

$=\frac{1}{3}\int \frac{dx}{{(x+\frac{13}{6})}^2-{\left(\frac{17}{6}\right)}^2}$

Using formula

$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\text{log}\left|\frac{x-a}{x+a}\right|+C$

$=\frac{1}{3}\times \frac{1}{2\left(\frac{17}{6}\right)}\times \text{log}\left|\frac{x+\frac{13}{6}-\frac{17}{6}}{x+\frac{13}{6}+\frac{17}{6}}\right|+C$

Where $C$ is constant of integration.

$=\frac{1}{17}\times \text{log}\left|\frac{x-\frac{4}{6}}{x+\frac{30}{6}}\right|+C$

$=\frac{1}{17}\times \text{log}\left|\frac{x-\frac{2}{3}}{x+5}\right|+C$

$=\frac{1}{17}\times \text{log}\left|\frac{3x-2}{3(x+5)}\right|+C$

​​​​​​​$=\frac{1}{17}\times \text{log}\left|\frac{3x-2}{x+5}\right|-\frac{1}{17}\text{log}3+C$

​​​​​​​$=\frac{1}{17}\times \text{log}\left|\frac{3x-2}{x+5}\right|+C_1$

Where $C_1$​​​​​​​$=C-\frac{1}{17}\text{log}3$


(iii) $\int \frac{dx}{\sqrt{5x^2-2x}}$

$=\int \frac{dx}{\sqrt{5(x^2-\frac{2}{5}x)}}$

$=\frac{1}{\sqrt{5}}\int \frac{dx}{\sqrt{x^2-\frac{2}{5}x}}$

$=\frac{1}{\sqrt{5}}\int \frac{dx}{\sqrt{x^2-\frac{2}{5}x+{\left(\frac{1}{5}\right)}^2-{\left(\frac{1}{5}\right)}^2}}$

$=\frac{1}{\sqrt{5}}\int \frac{dx}{\sqrt{{(x-\frac{1}{5})}^2-{\left(\frac{1}{5}\right)}^2}}$

$=\frac{1}{\sqrt{5}}\text{log}|(x-\frac{1}{5})+\sqrt{{(x-\frac{1}{5})}^2-{\left(\frac{1}{5}\right)}^2}|+C$

$[\because \int \frac{dx}{\sqrt{x^2-a^2}}=\text{log}|x+\sqrt{x^2-a^2}|+C]$

$=\frac{1}{\sqrt{5}}\text{log}|x-\frac{1}{5}+\sqrt{x^2+{\left(\frac{1}{5}\right)}^2-2\left(x\right)\left(\frac{1}{5}\right)-{\left(\frac{1}{5}\right)}^2}|+C$

$=\frac{1}{\sqrt{5}}\text{log}|x-\frac{1}{5}+\sqrt{x^2-\frac{2x}{5}}|+C$

Where $C$ is constant of integration