What is the integral from 0 to 17 of 2pix(17x-x^2) ?

The integral from 0 to 17 of 2pix(17x-x^2) is (83521pi)/6

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integral from 0 to 17 of 2pix(17x-x^2)

\int_{0}^{17} 2 π x (17 x - x^{2}) d x

\frac{83521 π}{6}

Decimal

43731.49333 \dots

Solution steps

\int_{0}^{17} 2 π x (17 x - x^{2}) d x

Take the constant out:

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

= 2 π \cdot \int_{0}^{17} x (17 x - x^{2}) d x

Expand

x (17 x - x^{2}) : 17 x^{2} - x^{3}

= 2 π \cdot \int_{0}^{17} 17 x^{2} - x^{3} 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 π (\int_{0}^{17} 17 x^{2} d x - \int_{0}^{17} x^{3} d x)

\int_{0}^{17} 17 x^{2} d x = \frac{83521}{3}

\int_{0}^{17} x^{3} d x = \frac{83521}{4}

= 2 π (\frac{83521}{3} - \frac{83521}{4})

Simplify

2 π (\frac{83521}{3} - \frac{83521}{4}) : \frac{83521 π}{6}

= \frac{83521 π}{6}

x \to a lim (\frac{sec ( x ) - sec ( a )}{x - a})

\int (\frac{7}{x} + 7 x) d x

y = 6 - ln (x + 1), y = ln (x + 1), x = 0

\int \frac{a}{( a ^{2} + x ^{2} ) ^{\frac{3}{2}}} d x

\int_{3}^{6} \frac{x}{3 x ^{2} - 8} d x

What is the integral from 0 to 17 of 2pix(17x-x^2) ?
The integral from 0 to 17 of 2pix(17x-x^2) is (83521pi)/6