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1 ∫ − 1 Log ( 2 − X 2 + X ) D X

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Question

\[\int\limits_{- 1}^1 \log\left( \frac{2 - x}{2 + x} \right) dx\]

Sum

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Solution

\[Let\ I = \int_{- 1}^1 \log\frac{2 - x}{2 + x} d x\]
\[Here\ f\left( x \right) = \log\left( \frac{2 - x}{2 + x} \right)\]
\[f\left( - x \right) = \log\left( \frac{2 + x}{2 - x} \right)\]
\[ = - \log\left( \frac{2 - x}{2 + x} \right)\]
\[ = - f\left( x \right)\]
\[\text{Hence} f\left( x \right) \text{is an odd function}, \]
Therefore,
\[I = 0\]

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Definite Integrals

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Chapter 19: Definite Integrals - Exercise 20.5 [Page 95]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12

Chapter 19 Definite Integrals
Exercise 20.5 | Q 25 | Page 95

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