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 2(1/4 ln(3)-pi/(12sqrt(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

2 (\frac{1}{4} ln (3) - \frac{π}{12 3})

Decimal Notation

0.24700 \dots

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

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

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

\int_{\frac{1}{2}}^{\frac{3}{2}} \frac{1}{4 u ^{2} + 3} d u = \frac{π}{12 3}

= 2 (\frac{1}{4} ln (3) - \frac{π}{12 3})

\frac{d x}{d t} + 5 t^{7} x^{9} + \frac{x}{t} = 0

x \to 0 - lim (\frac{ln ( x ^{2} )}{x})

\frac{x ^{2}}{1 - 2 x}

2 x^{^{''}} + 7 x^{^{'}} + 3 x = 0

\frac{d}{d x} (x (2 x + 7)^{\frac{1}{3}})

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 2(1/4 ln(3)-pi/(12sqrt(3)))