Which of the following is a correct substitution for the integral $\int _ { 1 } ^ { e } \frac { \cos ( \ln x ) } { x } d x$ ?
A) u = ln x; $\int _ { 0 } ^ { 1 } \sin u d u$
B) u = $\frac { \ln x } { x }$ ; $\int _ { 1 } ^ { \mathrm { e } } \cos \mathrm { u } d u$
C) u = ln x; $\int _ { 1 } ^ { \mathrm { e } } \cos \mathrm { u } d u$
D) u= ln x; $\int _ { 0 } ^ { 1 }$ $$\cos u$$
U du
E) u = ln x; $\int _ { 0 } ^ { 1 } \cos u \mathrm { du }$

Question

Quizplus · Accepted Answer

The correct substitution is $u = \ln x$, which transforms the integral bounds from $x = 1$ to $x = e$ into $u = 0$ to $u = 1$, and the integral becomes $\int _ { 0 } ^ { 1 } \cos u \, du$, since $dx/x = du$.

Which of the Following Is a Correct Substitution for the Integral