3. Define In(x) = f du. Fix x € (-1, 1). Use u-substitution to prove that (1+2). Hint: Use X [.—₁ - -X x dt = ln 1 1²₁₁²²₁d²=1²₁ ₁²² ₁² dt - t 1 t --- -S dt + C2 1 1 wwwww t dt.
3. Define In(x) = f du. Fix x € (-1, 1). Use u-substitution to prove that (1+2). Hint: Use X [.—₁ - -X x dt = ln 1 1²₁₁²²₁d²=1²₁ ₁²² ₁² dt - t 1 t --- -S dt + C2 1 1 wwwww t dt.
3. Define In(x) = f du. Fix x € (-1, 1). Use u-substitution to prove that (1+2). Hint: Use X [.—₁ - -X x dt = ln 1 1²₁₁²²₁d²=1²₁ ₁²² ₁² dt - t 1 t --- -S dt + C2 1 1 wwwww t dt.
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
\[
\int_{-x}^x \frac{1}{1-t} \, dt = \int_{-x}^0 \frac{1}{1-t} \, dt + \int_0^x \frac{1}{1-t} \, dt.
\]
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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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.
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