Question

Q 6 7 8 6.5CHOOSE ONE 6 7 OR 86 7 OR 8

Answer 1

Step-by-step explanation

(6) We must solve the following integral:

`I=\int_1^(e^7)dx\cdot\frac{(ln\text{ }x)^3}{x}.`

1) We make the following change of variables:

`\begin{gathered} u=\ln(u)\Rightarrow du=(dx)/(x), \\ u_2=\ln(e^7)=7, \\ u_1=\ln(1)=0. \end{gathered}`

With this change of variables, the integral is:

`I=\int_0^7du\cdot u^3.`

2) The result of the integral in terms of u is:

`I=(1)/(4)*u^4|_0^7=(1)/(4)*(7^4-0^4)=(2401)/(4)=600.25`Answer

600.25