3. Define ln(x) = fidu. Fix x € (-1,1). Use u-substitution to prove that Hint: Use 4 Use the Corall x -X [ ²₁ — —_-, dt = in (1 + ²). ln t 1 7 dt = 1²₁² t T 1 dt + 1-t CE 1 dt.
3. Define ln(x) = fidu. Fix x € (-1,1). Use u-substitution to prove that Hint: Use 4 Use the Corall x -X [ ²₁ — —_-, dt = in (1 + ²). ln t 1 7 dt = 1²₁² t T 1 dt + 1-t CE 1 dt.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. Define \(\ln(x) = \int_1^x \frac{1}{u} \, du\). Fix \(x \in (-1, 1)\). Use \(u\)-substitution to prove that
\[
\int_{-x}^x \frac{1}{1-t} \, dt = \ln \left( \frac{1+x}{1-x} \right).
\]
Hint: Use
\[
\int_{-x}^x \frac{1}{1-t} \, dt = \int_{-x}^0 \frac{1}{1-t} \, dt + \int_0^x \frac{1}{1-t} \, dt.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79599c56-a340-49a0-b0ff-829b3947a798%2F391f9e60-4958-44eb-82e8-8c1e09e97de8%2Fv2b1xo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Define \(\ln(x) = \int_1^x \frac{1}{u} \, du\). Fix \(x \in (-1, 1)\). Use \(u\)-substitution to prove that
\[
\int_{-x}^x \frac{1}{1-t} \, dt = \ln \left( \frac{1+x}{1-x} \right).
\]
Hint: Use
\[
\int_{-x}^x \frac{1}{1-t} \, dt = \int_{-x}^0 \frac{1}{1-t} \, dt + \int_0^x \frac{1}{1-t} \, dt.
\]
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