Explanation
(6) We must solve the following integral:
[tex]I=\int_1^{e^7}dx\cdot\frac{(ln\text{ }x)^3}{x}.[/tex]
1) We make the following change of variables:
[tex]\begin{gathered} u=\ln(u)\Rightarrow du=\frac{dx}{x}, \\ u_2=\ln(e^7)=7, \\ u_1=\ln(1)=0. \end{gathered}[/tex]
With this change of variables, the integral is:
[tex]I=\int_0^7du\cdot u^3.[/tex]
2) The result of the integral in terms of u is:
[tex]I=\frac{1}{4}*u^4|_0^7=\frac{1}{4}*(7^4-0^4)=\frac{2401}{4}=600.25[/tex]Answer
600.25
