What is the integral from 1 to 4 of pi(x^2-4x+5)^2 ?

The integral from 1 to 4 of pi(x^2-4x+5)^2 is pi 78/5

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integral from 1 to 4 of pi(x^2-4x+5)^2

\int_{1}^{4} π (x^{2} - 4 x + 5)^{2} d x

π \frac{78}{5}

Decimal

49.00884 \dots

Solution steps

\int_{1}^{4} π (x^{2} - 4 x + 5)^{2} d x

Take the constant out:

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

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

Expand

(x^{2} - 4 x + 5)^{2} : x^{4} - 8 x^{3} + 26 x^{2} - 40 x + 25

= π \cdot \int_{1}^{4} x^{4} - 8 x^{3} + 26 x^{2} - 40 x + 25 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

= π (\int_{1}^{4} x^{4} d x - \int_{1}^{4} 8 x^{3} d x + \int_{1}^{4} 26 x^{2} d x - \int_{1}^{4} 40 x d x + \int_{1}^{4} 25 d x)

\int_{1}^{4} x^{4} d x = \frac{1023}{5}

\int_{1}^{4} 8 x^{3} d x = 510

\int_{1}^{4} 26 x^{2} d x = 546

\int_{1}^{4} 40 x d x = 300

\int_{1}^{4} 25 d x = 75

= π (\frac{1023}{5} - 510 + 546 - 300 + 75)

Simplify

π (\frac{1023}{5} - 510 + 546 - 300 + 75) : π \frac{78}{5}

= π \frac{78}{5}

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What is the integral from 1 to 4 of pi(x^2-4x+5)^2 ?
The integral from 1 to 4 of pi(x^2-4x+5)^2 is pi 78/5