Question
\( \leftarrow \int_{1}^{\text {Evaluate the integral. }} \) \( \int_{1}^{6} \frac{4(\ln x)^{3}}{x} d x \) \( \int_{1}^{6} \frac{4(\ln x)^{3}}{x} d x=\square \) (Type an exact answer.)
Ask by O'Quinn Henry. in the United States
Mar 07,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral evaluates to \( (\ln 6)^4 \).
Solution
We start with the integral
\[
\int_{1}^{6} \frac{4(\ln x)^{3}}{x} \, dx.
\]
**Step 1.** Use the substitution
\[
u = \ln x \quad \Rightarrow \quad du = \frac{1}{x} \, dx.
\]
**Step 2.** Change the limits of integration:
- When \( x = 1 \), \( u = \ln 1 = 0 \).
- When \( x = 6 \), \( u = \ln 6 \).
**Step 3.** Substitute into the integral:
\[
\int_{1}^{6} \frac{4(\ln x)^{3}}{x} \, dx = 4 \int_{0}^{\ln 6} u^3 \, du.
\]
**Step 4.** Evaluate the integral:
\[
\int u^3 \, du = \frac{u^4}{4} + C.
\]
Thus,
\[
4 \int_{0}^{\ln 6} u^3 \, du = 4 \left[ \frac{u^4}{4} \right]_{0}^{\ln 6} = \left[ u^4 \right]_{0}^{\ln 6} = (\ln 6)^4 - 0.
\]
**Final Answer:**
\[
(\ln 6)^4.
\]
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Simplify this solution Extra Insights
To evaluate the integral \( \int_{1}^{6} \frac{4(\ln x)^{3}}{x} d x \), we can use substitution. Let \( u = \ln x \). Then, \( du = \frac{1}{x} dx \), or \( dx = x \, du = e^{u} \, du \). When \( x = 1 \), \( u = \ln 1 = 0 \), and when \( x = 6 \), \( u = \ln 6 \). The integral becomes: \[ \int_{0}^{\ln 6} 4u^{3} \, du. \] Now, we can find the integral \( \int 4u^{3} \, du \): \[ = 4 \cdot \frac{u^{4}}{4} = u^{4}. \] Evaluating this from \( 0 \) to \( \ln 6 \): \[ = (\ln 6)^{4} - (0)^{4} = (\ln 6)^{4}. \] Thus, the final answer is: \[ \int_{1}^{6} \frac{4(\ln x)^{3}}{x} d x = (\ln 6)^{4}. \]
