To evaluate the integral

$\int_{e^4}^{e} \frac{(\ln x)^3}{x} \, dx$

we'll use a substitution method to simplify the integrand.

Step 1: Substitute

Let $u = \ln x$. Then, $\frac{du}{dx} = \frac{1}{x}$, which implies $dx = x \, du = e^u \, du$.

Now, when $x = e^4$, $u = \ln(e^4) = 4$, and when $x = e$, $u = \ln(e) = 1$. This changes the bounds to $u$ ranging from $4$ to $1$.

Step 2: Rewrite the Integral

The integrand $\frac{(\ln x)^3}{x}$ in terms of $u$ becomes: $\frac{(\ln x)^3}{x} \, dx = u^3 \, du.$

Thus, the integral becomes: $\int_{e^4}^{e} \frac{(\ln x)^3}{x} \, dx = \int_4^1 u^3 \, du.$

Step 3: Integrate

Now we compute $\int u^3 \, du$: $\int u^3 \, du = \frac{u^4}{4}.$

Step 4: Evaluate at the Bounds

Evaluating from $u = 4$ to $u = 1$: $\int_4^1 u^3 \, du = \left[ \frac{u^4}{4} \right]_4^1 = \frac{1^4}{4} - \frac{4^4}{4}.$

Calculating each term: $= \frac{1}{4} - \frac{256}{4} = \frac{1}{4} - 64 = -\frac{255}{4}.$

Final Answer

$\int_{e^4}^{e} \frac{(\ln x)^3}{x} \, dx = -\frac{255}{4}.$

Would you like further explanation on any step?

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Related Questions:

1. How do we apply substitution to simplify different integrals?
2. How can we approach integration when the integrand includes logarithmic terms?
3. What other techniques are useful when evaluating definite integrals?
4. How does changing the order of integration bounds affect the final answer?
5. Can integrals involving $\ln x$ be solved using parts instead of substitution?

Tip: Always check if substitution changes the bounds; this can prevent mistakes in definite integrals.