int e^{x}left(frac{1-x}{1+x^{2}}right)^{2} ,d | vedclass

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(A) We know that $\int e^{x}[f(x) + f'(x)] dx = e^{x}f(x) + C$.Given integral is $I = \int e^{x} \left( \frac{1-x}{1+x^{2}} \right)^{2} dx$.Expanding

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$e^{x}\left(\frac{1}{1+x^{2}}\right)+C$

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(A) We know that $\int e^{x}[f(x) + f'(x)] dx = e^{x}f(x) + C$.Given integral is $I = \int e^{x} \left( \frac{1-x}{1+x^{2}} \right)^{2} dx$.Expanding the numerator: $I = \int e^{x} \frac{1 + x^{2} - 2x}{(1+x^{2})^{2}} dx$.Splitting the fraction: $I = \int e^{x} \left[ \frac{1+x^{2}}{(1+x^{2})^{2}} - \frac{2x}{(1+x^{2})^{2}} \right] dx$.$I = \int e^{x} \left[ \frac{1}{1+x^{2}} - \frac{2x}{(1+x^{2})^{2}} \right] dx$.Let $f(x) = \frac{1}{1+x^{2}}$. Then $f'(x) = -\frac{1}{(1+x^{2})^{2}} \cdot (2x) = -\frac{2x}{(1+x^{2})^{2}}$.Thus,$I = \int e^{x} [f(x) + f'(x)] dx = e^{x} f(x) + C$.Therefore,$I = \frac{e^{x}}{1+x^{2}} + C$.

$\int e^{x}\left(\frac{1-x}{1+x^{2}}\right)^{2} \,d x=$

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