Trigonometric substitution

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[tex]
\int\frac{x}{\sqrt{x^2+x+1}}dx
[/tex]
[tex]
\int \frac{x}{\sqrt{(x+\frac{1}{2})^2+\frac{3}{4}}}dx
[/tex]
[tex]
u=x+\frac{1}{2}
[/tex]
[tex]
\int \frac{u-\frac{1}{2}}{\sqrt{u^2+\frac{3}{4}}}du
[/tex]
[tex]
u=\frac{\sqrt{3}}{2}tanT
[/tex]
[tex]
du=\frac{\sqrt{3}}{2}sec^2TdT
[/tex]
[tex]
\int \frac{\frac{(\sqrt{3}}{2}tanT-\frac{1}{2})\frac{\sqrt{3}}{2}sec^2T}{\frac{\sqrt{3}}{2}secT}dT
[/tex]
[tex]
\int (\frac{\sqrt{3}}{2}tanT-\frac{1}{2})secTdT
[/tex]
[tex]
\frac{\sqrt{3}}{2}secT-\frac{1}{2}ln|secT+tanT|+C
[/tex]
[tex]
\frac{\sqrt{u^2+\frac{3}{4}}}{2}-\frac{ln|\sqrt{u^2+\frac{3}{4}}+2u|}{2}+C
[/tex]
[tex]
\frac{\sqrt{x^2+x+1}}{2}-\frac{ln|\sqrt{x^2+x+1}-2x+1|}{2}
[/tex]

Homework Statement





Homework Equations





The Attempt at a Solution

 

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