What is the integral from 0 to 1 of x/((x+1)^8) ?

The integral from 0 to 1 of x/((x+1)^8) is 5/224

integral from 0 to 1 of x/((x+1)^8)

\int_{0}^{1} \frac{x}{( x + 1 ) ^{8}} d x

\frac{5}{224}

Decimal

0.02232 \dots

Solution steps

\int_{0}^{1} \frac{x}{( x + 1 ) ^{8}} d x

Apply u-substitution

= \int_{1}^{2} \frac{u - 1}{u ^{8}} d u

Expand

\frac{u - 1}{u ^{8}} : \frac{1}{u ^{7}} - \frac{1}{u ^{8}}

= \int_{1}^{2} \frac{1}{u ^{7}} - \frac{1}{u ^{8}} d u

Apply the Sum Rule:

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

= \int_{1}^{2} \frac{1}{u ^{7}} d u - \int_{1}^{2} \frac{1}{u ^{8}} d u

\int_{1}^{2} \frac{1}{u ^{7}} d u = \frac{21}{128}

\int_{1}^{2} \frac{1}{u ^{8}} d u = \frac{127}{896}

= \frac{21}{128} - \frac{127}{896}

Simplify

\frac{21}{128} - \frac{127}{896} : \frac{5}{224}

= \frac{5}{224}

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What is the integral from 0 to 1 of x/((x+1)^8) ?
The integral from 0 to 1 of x/((x+1)^8) is 5/224