What is the integral from 1 to 2 of (2x^3-1)^26x^2 ?

The integral from 1 to 2 of (2x^3-1)^26x^2 is 3374/3

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integral from 1 to 2 of (2x^3-1)^26x^2

\int_{1}^{2} (2 x^{3} - 1)^{2} 6 x^{2} d x

\frac{3374}{3}

Decimal

1124.66666 \dots

Solution steps

\int_{1}^{2} (2 x^{3} - 1)^{2} \cdot 6 x^{2} d x

Take the constant out:

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

= 6 \cdot \int_{1}^{2} (2 x^{3} - 1)^{2} x^{2} d x

Expand

(2 x^{3} - 1)^{2} x^{2} : 4 x^{8} - 4 x^{5} + x^{2}

= 6 \cdot \int_{1}^{2} 4 x^{8} - 4 x^{5} + 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

= 6 (\int_{1}^{2} 4 x^{8} d x - \int_{1}^{2} 4 x^{5} d x + \int_{1}^{2} x^{2} d x)

\int_{1}^{2} 4 x^{8} d x = \frac{2044}{9}

\int_{1}^{2} 4 x^{5} d x = 42

\int_{1}^{2} x^{2} d x = \frac{7}{3}

= 6 (\frac{2044}{9} - 42 + \frac{7}{3})

Simplify

6 (\frac{2044}{9} - 42 + \frac{7}{3}) : \frac{3374}{3}

= \frac{3374}{3}

What is the integral from 1 to 2 of (2x^3-1)^26x^2 ?
The integral from 1 to 2 of (2x^3-1)^26x^2 is 3374/3