evaluate the definite integral.\n int_{1}^{4}\frac{12(ln x)^{3}}{x}dx \n int_{1}^{4}\frac{12(ln…

Explanation:

Step1: Use substitution

Let $u = \ln x$, then $du=\frac{1}{x}dx$. When $x = 1$, $u=\ln(1)=0$; when $x = 4$, $u=\ln(4)$. The integral $\int_{1}^{4}\frac{12(\ln x)^{3}}{x}dx$ becomes $12\int_{0}^{\ln(4)}u^{3}du$.

Step2: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). So, $12\int_{0}^{\ln(4)}u^{3}du=12\times\left[\frac{u^{4}}{4}\right]_{0}^{\ln(4)}$.

Step3: Evaluate the definite integral

$12\times\left[\frac{u^{4}}{4}\right]{0}^{\ln(4)} = 3\left[u^{4}\right]{0}^{\ln(4)}=3((\ln(4))^{4}-0^{4})=3(\ln(4))^{4}$.

Step4: Calculate the numerical value

$3(\ln(4))^{4}\approx3\times(1.386294)^{4}\approx3\times3.643=10.929$.

Answer:

$10.929$