#8747; #160;x2+x+1 #160;dx= #8747; #160;x2+x+122 122+1 #160;dx= #8747; #160;x+122+322=x+122 #160;x+122+322+38log #160;x+12+12+x2+x+1+C #1 ...

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Substitution Method to Remove Indeterminate Form

∫ x 2+x+1 d x

Question

$\int \sqrt{x^{2} + x + 1} d x$

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Solution

$\int \sqrt{x^{2} + x + 1} d x = \int \sqrt{x^{2} + x + {(\frac{1}{2})}^{2} - {(\frac{1}{2})}^{2} + 1} d x = \int \sqrt{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} = (\frac{x + \frac{1}{2}}{2}) \sqrt{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} + \frac{3}{8} \log |(x + \frac{1}{2}) + \frac{1}{2} + \sqrt{x^{2} + x + 1}| + C [∵ \int \sqrt{x^{2} + a^{2}} d x = \frac{1}{2} x \sqrt{x^{2} + a^{2}} + \frac{1}{2} a^{2} \ln |x + \sqrt{x^{2} + a^{2}}| + C] = (\frac{2 x + 1}{4}) \sqrt{x^{2} + x + 1} + \frac{3}{8} \log |(2 x + 1) + \frac{1}{2} + \sqrt{x^{2} + x + 1}| + C$

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