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  4. 37. If ∫ ex c ⋅ log (x2+1)+(b x/x2+1) d x=(b c/2) ex ⋅ log (x2+1) then (b, c) is

Q. 37. If $\int e^{x}\left\{c \cdot \log \left(x^{2}+1\right)+\frac{b x}{x^{2}+1}\right\} d x=\frac{b c}{2} e^{x} \cdot \log \left(x^{2}+1\right)$ then $(b, c)$ is

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Solution:

Derivative of $\log \left(x^{2}+1\right)$ is $\frac{2 x}{x^{2}+1}$.
$\therefore \int e^{x}\left\{\log \left(x^{2}+1\right)+\frac{2 x}{x^{2}+1}\right\} d x$
$=e^{x} \cdot \log \left(x^{2}+1\right) $
$\therefore(b, c) \equiv(2,1)$

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