Evaluate the following definite integral: `int_0^1 1/(2x^2+x+1)dx`

Explanation

Denominator is a quadratic equation So, make it in form of a perfect square

`2 x^{2}+x+1=(x^{2}+frac{x}{2}+frac{1}{2}) times 2 `
`
=((x+frac{1}{4})^{2}+frac{1}{2}-frac{1}{16}) times 2 `
`
=((x+frac{1}{4})^{2}+frac{7}{16}) times 2 `
`
=((x+frac{1}{4})^{2}+(frac{sqrt{7}}{4})^{2}) times 2`


We get the equation
`
int_{0}^{1} frac{1}{2 x^{2}+x+1}=int_{0}^{1} frac{1}{((x+frac{1}{4})^{2}+(frac{sqrt{7}}{4})^{2}) times 2}`
`
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