int left( frac{(x^2+2) a^{(x+tan^{-1} x)}}{x^ | vedclass

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Question

(C) Let $I = \int \frac{(x^2+2) a^{(x+	an^{-1} x)}}{x^2+1} dx$. We can rewrite the integrand as: $I = \int \left( \frac{x^2+1+1}{x^2+1} ight) a^{(x

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$\frac{a^{x+	an^{-1} x}}{\log a}+c$

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(C) Let $I = \int \frac{(x^2+2) a^{(x+	an^{-1} x)}}{x^2+1} dx$. We can rewrite the integrand as: $I = \int \left( \frac{x^2+1+1}{x^2+1} ight) a^{(x+	an^{-1} x)} dx = \int \left( 1 + \frac{1}{x^2+1} ight) a^{(x+	an^{-1} x)} dx$. Let $u = x + 	an^{-1} x$. Then,$du = (1 + \frac{1}{1+x^2}) dx$. Substituting these into the integral: $I = \int a^u du$. Using the standard integral formula $\int a^u du = \frac{a^u}{\ln a} + C$,we get: $I = \frac{a^u}{\ln a} + C$. Substituting back $u = x + 	an^{-1} x$: $I = \frac{a^{x+	an^{-1} x}}{\ln a} + C$.

$\int \left( \frac{(x^2+2) a^{(x+\tan^{-1} x)}}{x^2+1} \right) dx = $ . . . . . .

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