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问题:求积分∫(x+1)/√(x^2+x+1)dx

求积分∫(x+1)/√(x^2+x+1)dx

解答

求积分 \int\frac{x+1}{\sqrt{x^2+x+1}}dx

\begin{aligned}
&\int \frac{x+1}{\sqrt{x^{2}+x+1}} dx\\
=& \int \frac{x+1}{\sqrt{\left(x+\frac{1}{2}\right)^{2}+\frac{3}{4}}} d x\\
=& \frac{2}{\sqrt{3}} \int \frac{x+1}{\sqrt{\frac{(2 x+1)^{2}}{3}+1}} d x\\
=& \frac{2}{\sqrt{3}} \int \frac{\frac{\sqrt{3} u-1}{2}+1}{\sqrt{u^{2}+1}} d \frac{\sqrt{3} u-1}{2} \cdot\left(u=\frac{2 x+1}{\sqrt{3}}, x=\frac{\sqrt{3} u-1}{2}\right)\\
=& \frac{1}{2} \int \frac{\sqrt3 u+1}{\sqrt{u^{2}+1}} d u\\
=& \frac{\sqrt 3}{2} \int \frac{u}{\sqrt{u^{2}+1}} d u+\frac{1}{2} \int \frac{1}{\sqrt{u^{2}+1}} d u\\
=& \sqrt{x^{2}+x+1} +\frac{1}{2} arcsinh \frac{\sqrt{3}}{3}(2 x+1)+C\\
\end{aligned}

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