What is the integral from-3 to 0 of 3x^2(3x^3+3)^6 ?

The integral from-3 to 0 of 3x^2(3x^3+3)^6 is 3((3^{20}*729)/7-84305136474)

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integral from-3 to 0 of 3x^2(3x^3+3)^6

\int_{- 3}^{0} 3 x^{2} (3 x^{3} + 3)^{6} d x

3 (\frac{3 ^{20} \cdot 729}{7} - 84305136474)

Decimal Notation

836455659861.8572

Solution steps

\int_{- 3}^{0} 3 x^{2} (3 x^{3} + 3)^{6} d x

Take the constant out:

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

= 3 \cdot \int_{- 3}^{0} x^{2} (3 x^{3} + 3)^{6} d x

Expand

x^{2} (3 x^{3} + 3)^{6} : 729 x^{20} + 4374 x^{17} + 10935 x^{14} + 14580 x^{11} + 10935 x^{8} + 4374 x^{5} + 729 x^{2}

= 3 \cdot \int_{- 3}^{0} 729 x^{20} + 4374 x^{17} + 10935 x^{14} + 14580 x^{11} + 10935 x^{8} + 4374 x^{5} + 729 x^{2} 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_{- 3}^{0} 729 x^{20} d x + \int_{- 3}^{0} 4374 x^{17} d x + \int_{- 3}^{0} 10935 x^{14} d x + \int_{- 3}^{0} 14580 x^{11} d x + \int_{- 3}^{0} 10935 x^{8} d x + \int_{- 3}^{0} 4374 x^{5} d x + \int_{- 3}^{0} 729 x^{2} d x)

\int_{- 3}^{0} 729 x^{20} d x = \frac{3 ^{20} \cdot 729}{7}

\int_{- 3}^{0} 4374 x^{17} d x = - 94143178827

\int_{- 3}^{0} 10935 x^{14} d x = 10460353203

\int_{- 3}^{0} 14580 x^{11} d x = - 645700815

\int_{- 3}^{0} 10935 x^{8} d x = 23914845

\int_{- 3}^{0} 4374 x^{5} d x = - 531441

\int_{- 3}^{0} 729 x^{2} d x = 6561

= 3 (\frac{3 ^{20} \cdot 729}{7} - 94143178827 + 10460353203 - 645700815 + 23914845 - 531441 + 6561)

Simplify

= 3 (\frac{3 ^{20} \cdot 729}{7} - 84305136474)

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What is the integral from-3 to 0 of 3x^2(3x^3+3)^6 ?
The integral from-3 to 0 of 3x^2(3x^3+3)^6 is 3((3^{20}*729)/7-84305136474)