Mathematics - Definite Integration Question with Solution | TestHub

Let $I={\int }_1^e\frac{{\left(\ln x\right)}^{{\displaystyle \frac{1}{2}}}dx}{x{\left(a-{\left(\ln x\right)}^{{\displaystyle \frac{3}{2}}}\right)}^2}$

Now put $a-{\left(\ln x\right)}^{\frac{3}{2}}=t$

$\Rightarrow -\frac{3}{2}{\left(\ln x\right)}^{\frac{1}{2}}·\frac{1}{x}dx=dt$

So, the integral becomes $I={\int }_a^{a-1}\frac{\left(-\frac{2}{3}\right)dt}{t^2}$

$\Rightarrow I=\left(-\frac{2}{3}\right){\overline{)\frac{t^{-2+1}}{-2+1}}}_a^{a-1}$

$\Rightarrow I={\overline{)\frac{2}{3t}}}_a^{a-1}$

$\Rightarrow I=\frac{2}{3}\left(\frac{1}{a-1}-\frac{1}{a}\right)$

Now given $I=\left(\frac{2}{3}\right)\frac{1}{a\left(a-1\right)}=1$

$\Rightarrow 2=3a^2-3a$

$\Rightarrow 3a^2-3a-2=0$

$\Rightarrow a=\frac{3\pm \sqrt{9-4\left(3\right)\left(-2\right)}}{6}$

So, $a=\frac{3+\sqrt{33}}{6},\frac{3-\sqrt{33}}{6}$