$\int_{1}^{4} \frac{\ln{x}}{\sqrt{x}} dx$ এর মান কত?

$\int_{1}^{4} \frac{\ln x}{\sqrt{x}} \mathrm{dx}=\int_{1}^{4} \frac{2 \ln \sqrt{\mathrm{x}}}{\sqrt{\mathrm{x}}} \mathrm{dx}$

ধরি, $\sqrt{\mathrm{x}}=\mathrm{z} \Rightarrow \frac{1}{2 \sqrt{\mathrm{x}}} \mathrm{dx}=\mathrm{dz} \Rightarrow \frac{1}{\sqrt{\mathrm{x}}} \mathrm{dx}=2 \mathrm{dz}$

or

$\int \frac{2 \ln \sqrt{x}}{\sqrt{x}} d x=\int 4 \ln z d z$

$\begin{array}{l} \quad=4[z \ln z-z]=4(\sqrt{x} \ln \sqrt{x}-\sqrt{x}) \\ \therefore \int_{1}^{4} \frac{2 \ln \sqrt{x}}{\sqrt{x}} d x=4[\sqrt{x} \ln \sqrt{x}-\sqrt{x}]_{1}^{4} \\ =4(2 \ln 2-2)-4(1 \ln 1-1) \\ =8 \ln 2-4 \end{array}$