What is the integral from 3 to 6 of 2(x-4)^2 ?

The integral from 3 to 6 of 2(x-4)^2 is 6

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integral from 3 to 6 of 2(x-4)^2

\int_{3}^{6} 2 (x - 4)^{2} d x

6

Solution steps

\int_{3}^{6} 2 (x - 4)^{2} d x

Take the constant out:

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

= 2 \cdot \int_{3}^{6} (x - 4)^{2} d x

Expand

(x - 4)^{2} : x^{2} - 8 x + 16

= 2 \cdot \int_{3}^{6} x^{2} - 8 x + 16 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

= 2 (\int_{3}^{6} x^{2} d x - \int_{3}^{6} 8 x d x + \int_{3}^{6} 16 d x)

\int_{3}^{6} x^{2} d x = 63

\int_{3}^{6} 8 x d x = 108

\int_{3}^{6} 16 d x = 48

= 2 (63 - 108 + 48)

Simplify

= 6

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What is the integral from 3 to 6 of 2(x-4)^2 ?
The integral from 3 to 6 of 2(x-4)^2 is 6