$$ int _ { e } ^ { e ^ { 3 } } \frac { d u } { u ln u } $$
$$ int _ { e } ^ { e ^ { 3 } } \frac { d u } { u ln u } $$
Answer
Explanation:
Step1: Substitution
Let ( t = \ln u), then (dt=\frac{1}{u}du). When (u = e), (t=\ln e = 1); when (u=e^{3}), (t=\ln e^{3}=3). The integral (\int_{e}^{e^{3}}\frac{du}{u\ln u}) becomes (\int_{1}^{3}\frac{dt}{t}).
Step2: Integrate
The antiderivative of (\frac{1}{t}) is (\ln|t|). Using the fundamental theorem of calculus (\int_{1}^{3}\frac{dt}{t}=\left[\ln t\right]_{1}^{3}).
Step3: Evaluate
(\left[\ln t\right]_{1}^{3}=\ln 3-\ln 1). Since (\ln 1 = 0), the result is (\ln 3).
Answer:
(\ln 3)