What is the integral from 5 to 10 of (x^2+1)^5 ?

The integral from 5 to 10 of (x^2+1)^5 is 35836907815/63+(10^{11}-48828125)/(11)

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integral from 5 to 10 of (x^2+1)^5

\int_{5}^{10} (x^{2} + 1)^{5} d x

\frac{35836907815}{63} + \frac{1 0 ^{11} - 48828125}{11}

Decimal

9655309977.041847

Solution steps

\int_{5}^{10} (x^{2} + 1)^{5} d x

Expand

(x^{2} + 1)^{5} : x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} + 1

= \int_{5}^{10} x^{10} + 5 x^{8} + 10 x^{6} + 10 x^{4} + 5 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_{5}^{10} x^{10} d x + \int_{5}^{10} 5 x^{8} d x + \int_{5}^{10} 10 x^{6} d x + \int_{5}^{10} 10 x^{4} d x + \int_{5}^{10} 5 x^{2} d x + \int_{5}^{10} 1 d x

\int_{5}^{10} x^{10} d x = \frac{1 0 ^{11} - 48828125}{11}

\int_{5}^{10} 5 x^{8} d x = \frac{4990234375}{9}

\int_{5}^{10} 10 x^{6} d x = \frac{99218750}{7}

\int_{5}^{10} 10 x^{4} d x = 193750

\int_{5}^{10} 5 x^{2} d x = \frac{4375}{3}

\int_{5}^{10} 1 d x = 5

= \frac{1 0 ^{11} - 48828125}{11} + \frac{4990234375}{9} + \frac{99218750}{7} + 193750 + \frac{4375}{3} + 5

Simplify

\frac{1 0 ^{11} - 48828125}{11} + \frac{4990234375}{9} + \frac{99218750}{7} + 193750 + \frac{4375}{3} + 5 : \frac{35836907815}{63} + \frac{1 0 ^{11} - 48828125}{11}

= \frac{35836907815}{63} + \frac{1 0 ^{11} - 48828125}{11}

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What is the integral from 5 to 10 of (x^2+1)^5 ?
The integral from 5 to 10 of (x^2+1)^5 is 35836907815/63+(10^{11}-48828125)/(11)