Practicing Success
The value of integral $\int \sqrt{4 x^2+9} d x$ is |
$\frac{x}{2} \sqrt{4 x^2+9}+\frac{9}{2} \log \left|2 x+\sqrt{4 x^2+9}\right|+C$ $\frac{x}{2} \sqrt{4 x^2+9}+\frac{3}{2} \log \left|2 x+\sqrt{4 x^2+9}\right|+C$ $2 x \sqrt{4 x^2+9}+\frac{9}{2} \log \left|2 x+\sqrt{4 x^2+9}\right|+C$ $=\frac{x}{2} \sqrt{4 x^2+9}+\frac{9}{4} \log [\left(2 x+\sqrt{4 x^2+9}\right)]+C$ |
$=\frac{x}{2} \sqrt{4 x^2+9}+\frac{9}{4} \log [\left(2 x+\sqrt{4 x^2+9}\right)]+C$ |
$I=\int \sqrt{4 x^2+9} d x$ $=\int \sqrt{(2 x)^2+3^2} d x$ $\int \sqrt{y^2+3^2} \frac{d y}{2}$ let y = 2x ......(1) ⇒ $d y=2 d x $ $\Rightarrow \frac{d y}{2}=d x$ so expanding $\frac{1}{2}[\frac{y}{2}\sqrt{y^2+3^2} + \frac{3^2}{2}log |y+\sqrt{y^2+3^2}|]+C$ since $|\int \sqrt{x^2+a^2} dx = \frac{x}{2} \sqrt{x^2+ a^2} + \frac{a^2}{2} log |x^2+9^2| + C|$ from (1) reapplying of x in expression $\frac{1}{2}[\frac{2x}{2} \sqrt{(2x)^2 + 3^2} + \frac{3^2}{2} log 4x + \sqrt{(2x)^2+3^2}]+ C$ $=\frac{x}{2} \sqrt{4 x^2+9}+\frac{9}{4} \log [\left(2 x+\sqrt{4 x^2+9}\right)]+C$ |