evaluate the definite integral.\n int_{1}^{4}\frac{12(ln x)^{5}}{x}dx \n int_{1}^{4}\frac{12(ln…

evaluate the definite integral.\n int_{1}^{4}\frac{12(ln x)^{5}}{x}dx \n int_{1}^{4}\frac{12(ln x)^{5}}{x}dx=square \n(round to three decimal places as needed.)

evaluate the definite integral.\n int_{1}^{4}\frac{12(ln x)^{5}}{x}dx \n int_{1}^{4}\frac{12(ln x)^{5}}{x}dx=square \n(round to three decimal places as needed.)

Answer

Explanation:

Step1: Use substitution method

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)^{5}}{x}dx) can be rewritten as (12\int_{0}^{\ln(4)}u^{5}du).

Step2: Apply integral formula

The integral of (u^{n}) with respect to (u) is (\frac{u^{n + 1}}{n+1}+C) ((n\neq - 1)). So, (12\int_{0}^{\ln(4)}u^{5}du=12\times\left[\frac{u^{6}}{6}\right]_{0}^{\ln(4)}).

Step3: Evaluate the definite - integral

[ \begin{align*} 12\times\left[\frac{u^{6}}{6}\right]{0}^{\ln(4)}&=2\left[u^{6}\right]{0}^{\ln(4)}\ &=2((\ln(4))^{6}-0^{6})\ &=2(\ln(4))^{6} \end{align*} ] Using a calculator, ((\ln(4))^{6}\approx6.5432), and (2(\ln(4))^{6}\approx13.086).

Answer:

(13.086)