- #1

DryRun

Gold Member

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**Homework Statement**

Find [tex]\int \frac{x}{\sqrt{2x^2-2x+1}}\,dx[/tex]

**The attempt at a solution**

First, i complete the square for the quadratic expression:

[tex]2x^2-2x+1=2((x-\frac{1}{2})^2+\frac{1}{4})[/tex]

[tex]\int \frac{x}{\sqrt{2x^2-2x+1}}\,dx=\int \frac{x}{\sqrt 2 \sqrt{(x-\frac{1}{2})^2+\frac{1}{4}}}\,dx[/tex]

Using substitution: Let [itex](x-\frac{1}{2})=\frac{1}{2}\sinh u[/itex]

I eventually get this expression, after integrating:

[tex]\frac{1}{2\sqrt 2}(\cosh (\sinh^{-1}(2x-1)+\sinh^{-1}(2x-1))[/tex]

But i don't know how to simplify this any further, without getting a complicated answer. The final answer is: [itex]\frac{\sinh^{-1}1}{\sqrt 2}[/itex]

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