$\int_1^2 \frac{1}{x^2} e^{-\frac{1}{x}} \, dx = $

Consider the equation $\int_1^e \frac{(\log_e x)^{1/2}}{x(a-(\log_e x)^{3/2})^2} dx = 1$,where $a \in (-\infty, 0) \cup (1, \infty)$. Which of the following statements is/are $TRUE$?

$\int_0^{\pi /4} {\frac{{\sin x + \cos x}}{{9 + 16\sin 2x}}\,dx = } $

Let $f(x) = \int\limits_0^x {{e^{ - {t^2}}}dt} $ for all $x > 0$. Then for all $x > 0$:

If ${I_n} = \int_0^\infty {{e^{ - x}}{x^{n - 1}}dx,} $ then $\int_0^\infty {{e^{ - \lambda x}}{x^{n - 1}}dx = } $

$\int \limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} d x$ is equal to