Mathematics - Definite Integration Question with Solution | TestHub

${\displaystyle \underset{1}{\overset{2}{\int }}\frac{dx}{{\left({\left(x-1\right)}^2+3\right)}^{\frac{3}{2}}}}$

Substituting$x-1=\sqrt{3}\tan \theta$

$\Rightarrow dx=\sqrt{3}{\sec }^2\theta ·d\theta$

Now$\underset{0}{\overset{\frac{\pi }{6}}{\int }}\frac{\sqrt{3}{\sec }^2\mathrm{\theta }d\mathrm{\theta }}{3\sqrt{3}{\sec }^3\mathrm{\theta }}$

$=\frac{1}{3}\underset{0}{\overset{\frac{\pi }{6}}{\int }}\cos \theta d\theta =\frac{1}{3}{\left[\sin \theta \right]}_0^{\frac{\pi }{6}}=\frac{1}{6}$

$\Rightarrow \frac{1}{6}=\frac{k}{k+5}=k+5=6k$

$=k=1$.