What is the integral from 0 to 1 of 3x^5(6+x^3)^5 ?

The integral from 0 to 1 of 3x^5(6+x^3)^5 is 46656/7

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integral from 0 to 1 of 3x^5(6+x^3)^5

\int_{0}^{1} 3 x^{5} (6 + x^{3})^{5} d x

\frac{46656}{7}

Decimal Notation

6665.14285 \dots

Solution steps

\int_{0}^{1} 3 x^{5} (6 + x^{3})^{5} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int_{0}^{1} x^{5} (6 + x^{3})^{5} d x

Expand

x^{5} (6 + x^{3})^{5} : 7776 x^{5} + 6480 x^{8} + 2160 x^{11} + 360 x^{14} + 30 x^{17} + x^{20}

= 3 \cdot \int_{0}^{1} 7776 x^{5} + 6480 x^{8} + 2160 x^{11} + 360 x^{14} + 30 x^{17} + x^{20} 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

= 3 (\int_{0}^{1} 7776 x^{5} d x + \int_{0}^{1} 6480 x^{8} d x + \int_{0}^{1} 2160 x^{11} d x + \int_{0}^{1} 360 x^{14} d x + \int_{0}^{1} 30 x^{17} d x + \int_{0}^{1} x^{20} d x)

\int_{0}^{1} 7776 x^{5} d x = 1296

\int_{0}^{1} 6480 x^{8} d x = 720

\int_{0}^{1} 2160 x^{11} d x = 180

\int_{0}^{1} 360 x^{14} d x = 24

\int_{0}^{1} 30 x^{17} d x = \frac{5}{3}

\int_{0}^{1} x^{20} d x = \frac{1}{21}

= 3 (1296 + 720 + 180 + 24 + \frac{5}{3} + \frac{1}{21})

Simplify

3 (1296 + 720 + 180 + 24 + \frac{5}{3} + \frac{1}{21}) : \frac{46656}{7}

= \frac{46656}{7}

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What is the integral from 0 to 1 of 3x^5(6+x^3)^5 ?
The integral from 0 to 1 of 3x^5(6+x^3)^5 is 46656/7