Use the Change of Variables Formula to evaluate the definite integral.
\[
\int_{1}^{e^{6}} \frac{6 \ln x}{x} d x
\]
(Use symbolic notation and fractions where needed.)
\[
\int_{1}^{e^{5}} \frac{6 \ln x}{x} d x=
\]
Answer
Final Answer: The definite integral is \(\boxed{108}\).
Step 1 ：Use the Change of Variables Formula to evaluate the definite integral.
Step 2 ：\[\int_{1}^{e^{6}} \frac{6 \ln x}{x} d x\]
Step 3 ：Let \( u = \ln x \), then \( x = e^u \) and \( dx = e^u du \). The limits of integration also change from \( [1, e^6] \) to \( [0, 6] \).
Step 4 ：The integral becomes \[\int_{0}^{6} 6u du\]
Step 5 ：Integrate to get \( 3u^2 \) evaluated from 0 to 6.
Step 6 ：The final result is \( 3*6^2 = 108 \).
Step 7 ：Final Answer: The definite integral is \(\boxed{108}\).
