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

Expert–verified

Hide Steps

Answer

Final Answer: The definite integral is $108$ .

Steps

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 $d x = e^{u} d u$ . The limits of integration also change from $[1, e^{6}]$ to $[0, 6]$ .

Step 4 ：The integral becomes $\int_{0}^{6} 6 u d u$

Step 5 ：Integrate to get $3 u^{2}$ evaluated from 0 to 6.

Step 6 ：The final result is $3 * 6^{2} = 108$ .

Step 7 ：Final Answer: The definite integral is $108$ .