What is the integral from 0 to 1 of x/(x^2-x+1) ?

The integral from 0 to 1 of x/(x^2-x+1) is pi/(3sqrt(3))

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integral from 0 to 1 of x/(x^2-x+1)

\int_{0}^{1} \frac{x}{x ^{2} - x + 1} d x

\frac{π}{3 3}

Solution steps

\int_{0}^{1} \frac{x}{x ^{2} - x + 1} d x

Complete the square

x^{2} - x + 1 : (x - \frac{1}{2})^{2} + \frac{3}{4}

= \int_{0}^{1} \frac{x}{( x - \frac{1}{2} ) ^{2} + \frac{3}{4}} d x

Apply u-substitution

: \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2 ( 2 u + 1 )}{4 u ^{2} + 3} d u

= \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2 ( 2 u + 1 )}{4 u ^{2} + 3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2 u + 1}{4 u ^{2} + 3} d u

Expand

\frac{2 u + 1}{4 u ^{2} + 3} : \frac{2 u}{4 u ^{2} + 3} + \frac{1}{4 u ^{2} + 3}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2 u}{4 u ^{2} + 3} d u + \int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{4 u ^{2} + 3} d u)

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{2 u}{4 u ^{2} + 3} d u = 0

\int_{- \frac{1}{2}}^{\frac{1}{2}} \frac{1}{4 u ^{2} + 3} d u = \frac{π}{6 3}

= 2 (0 + \frac{π}{6 3})

Simplify

2 (0 + \frac{π}{6 3}) : \frac{π}{3 3}

= \frac{π}{3 3}

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What is the integral from 0 to 1 of x/(x^2-x+1) ?
The integral from 0 to 1 of x/(x^2-x+1) is pi/(3sqrt(3))