What is the integral from 0 to 1 of x(x+1)^{10} ?

The integral from 0 to 1 of x(x+1)^{10} is 6827/44

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integral from 0 to 1 of x(x+1)^{10}

\int_{0}^{1} x (x + 1)^{10} d x

\frac{6827}{44}

Decimal

155.15909 \dots

Solution steps

\int_{0}^{1} x (x + 1)^{10} d x

Apply u-substitution

: \int_{1}^{2} (u - 1) u^{10} d u

= \int_{1}^{2} (u - 1) u^{10} d u

Expand

(u - 1) u^{10} : u^{11} - u^{10}

= \int_{1}^{2} u^{11} - u^{10} 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} u^{11} d u - \int_{1}^{2} u^{10} d u

\int_{1}^{2} u^{11} d u = \frac{1365}{4}

\int_{1}^{2} u^{10} d u = \frac{2047}{11}

= \frac{1365}{4} - \frac{2047}{11}

Simplify

\frac{1365}{4} - \frac{2047}{11} : \frac{6827}{44}

= \frac{6827}{44}

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What is the integral from 0 to 1 of x(x+1)^{10} ?
The integral from 0 to 1 of x(x+1)^{10} is 6827/44