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

The integral from 0 to 1 of x/((x+1)(x^2+1)) is (-2ln(2)+pi)/8

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

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

\frac{- 2 ln ( 2 ) + π}{8}

Decimal

0.21941 \dots

Solution steps

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

Take the partial fraction of

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

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

Apply the Sum Rule:

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

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

\int_{0}^{1} \frac{1}{2 ( x + 1 )} d x = \frac{1}{2} ln (2)

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

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

Simplify

- \frac{1}{2} ln (2) + \frac{1}{2} (\frac{1}{2} ln (2) + \frac{π}{4}) : \frac{- 2 ln ( 2 ) + π}{8}

= \frac{- 2 ln ( 2 ) + π}{8}

What is the integral from 0 to 1 of x/((x+1)(x^2+1)) ?
The integral from 0 to 1 of x/((x+1)(x^2+1)) is (-2ln(2)+pi)/8