int_{-5}^{5} log left(frac{7-x}{7+x}right) dx | vedclass

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Question

(B) Let $I = \int_{-5}^{5} \log \left(\frac{7-x}{7+x}\right) dx$. Define $f(x) = \log \left(\frac{7-x}{7+x}\right)$. Check if the function is even or

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(B) Let $I = \int_{-5}^{5} \log \left(\frac{7-x}{7+x}\right) dx$. Define $f(x) = \log \left(\frac{7-x}{7+x}\right)$. Check if the function is even or odd by calculating $f(-x)$: $f(-x) = \log \left(\frac{7-(-x)}{7+(-x)}\right) = \log \left(\frac{7+x}{7-x}\right)$. Using the property $\log \left(\frac{a}{b}\right) = -\log \left(\frac{b}{a}\right)$,we get: $f(-x) = -\log \left(\frac{7-x}{7+x}\right) = -f(x)$. Since $f(-x) = -f(x)$,the function $f(x)$ is an odd function. According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) dx = 0$. Therefore,$I = 0$.

$\int_{-5}^{5} \log \left(\frac{7-x}{7+x}\right) dx =$

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