evaluate the definite integral.\n int_{1}^{e^{8}} \frac{dx}{xsqrt{ln x}}

Explanation:

Step1: Use substitution

Let $u = \ln x$. Then $du=\frac{1}{x}dx$. When $x = 1$, $u=\ln(1)=0$. When $x = e^{8}$, $u=\ln(e^{8}) = 8$.

Step2: Rewrite the integral

The integral $\int_{1}^{e^{8}}\frac{dx}{x\sqrt{\ln x}}$ becomes $\int_{0}^{8}\frac{du}{\sqrt{u}}=\int_{0}^{8}u^{-\frac{1}{2}}du$.

Step3: Integrate using power - rule

The antiderivative of $u^{-\frac{1}{2}}$ is $\frac{u^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=2u^{\frac{1}{2}}+C$.

Step4: Evaluate the definite integral

$2u^{\frac{1}{2}}\big|_{0}^{8}=2\sqrt{8}-2\sqrt{0}=4\sqrt{2}-0 = 4\sqrt{2}$.

Answer:

$4\sqrt{2}$