Compute the deﬁnite integral

∫(e^4,e)( lnx)^3/xdx=      Preview Change entry mode

Give the exact value.

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.

Compute the definite integral ∫(e^4,e) ((ln x)^3 / x) dx. Give the exact value.

This solution explains how to evaluate the definite integral ∫(e^4,e) ((ln x)^3 / x) dx using substitution. By transforming the integrand and updating the bounds, students find the exact value, -255/4. This exercise covers integration techniques suitable for advanced high school and early college calculus students.

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