For x0, evaluate the integral: int left( fra | vedclass

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(D) Let $I = \int \left( \frac{\sqrt{1+x+x^2}}{1+x} + \frac{1}{2 \sqrt{1+x+x^2}} - \frac{1}{(1+x) \sqrt{1+x+x^2}} \right) dx$. Combine the first and t

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$\sqrt{x^2+x+1}+C$

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(D) Let $I = \int \left( \frac{\sqrt{1+x+x^2}}{1+x} + \frac{1}{2 \sqrt{1+x+x^2}} - \frac{1}{(1+x) \sqrt{1+x+x^2}} \right) dx$. Combine the first and third terms: $I = \int \left( \frac{1+x+x^2-1}{(1+x) \sqrt{1+x+x^2}} \right) dx + \int \frac{1}{2 \sqrt{1+x+x^2}} dx$. Simplify the numerator: $I = \int \frac{x(1+x)}{(1+x) \sqrt{1+x+x^2}} dx + \int \frac{1}{2 \sqrt{1+x+x^2}} dx = \int \frac{x}{\sqrt{1+x+x^2}} dx + \int \frac{1}{2 \sqrt{1+x+x^2}} dx$. Multiply and divide the first integral by $2$: $I = \int \frac{2x}{2 \sqrt{1+x+x^2}} dx + \int \frac{1}{2 \sqrt{1+x+x^2}} dx = \int \frac{2x+1-1}{2 \sqrt{1+x+x^2}} dx + \int \frac{1}{2 \sqrt{1+x+x^2}} dx$. Split the integral: $I = \int \frac{2x+1}{2 \sqrt{1+x+x^2}} dx - \int \frac{1}{2 \sqrt{1+x+x^2}} dx + \int \frac{1}{2 \sqrt{1+x+x^2}} dx = \int \frac{2x+1}{2 \sqrt{1+x+x^2}} dx$. Let $t = x^2+x+1$, then $dt = (2x+1) dx$. $I = \frac{1}{2} \int \frac{1}{\sqrt{t}} dt = \sqrt{t} + C = \sqrt{x^2+x+1} + C$.

For $x>0$, evaluate the integral: $\int \left( \frac{\sqrt{1+x+x^2}}{1+x} + \frac{1}{2 \sqrt{1+x+x^2}} - \frac{1}{(1+x) \sqrt{1+x+x^2}} \right) dx$.

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