What is the integral from 0 to 5 of x^2(x-5)^4 ?

The integral from 0 to 5 of x^2(x-5)^4 is 15625/21

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integral from 0 to 5 of x^2(x-5)^4

\int_{0}^{5} x^{2} (x - 5)^{4} d x

\frac{15625}{21}

Decimal Notation

744.04761 \dots

Solution steps

\int_{0}^{5} x^{2} (x - 5)^{4} d x

Expand

x^{2} (x - 5)^{4} : x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 625 x^{2}

= \int_{0}^{5} x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 625 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

= \int_{0}^{5} x^{6} d x - \int_{0}^{5} 20 x^{5} d x + \int_{0}^{5} 150 x^{4} d x - \int_{0}^{5} 500 x^{3} d x + \int_{0}^{5} 625 x^{2} d x

\int_{0}^{5} x^{6} d x = \frac{78125}{7}

\int_{0}^{5} 20 x^{5} d x = \frac{156250}{3}

\int_{0}^{5} 150 x^{4} d x = 93750

\int_{0}^{5} 500 x^{3} d x = 78125

\int_{0}^{5} 625 x^{2} d x = \frac{78125}{3}

= \frac{78125}{7} - \frac{156250}{3} + 93750 - 78125 + \frac{78125}{3}

Simplify

\frac{78125}{7} - \frac{156250}{3} + 93750 - 78125 + \frac{78125}{3} : \frac{15625}{21}

= \frac{15625}{21}

x \to 2 lim (\frac{x - 2}{x ^{3} - 8})

f^{^{'}} (x) = x^{3} - 5 x^{2} + 3

256 = 4 x^{2} + 2 x y - 4 y^{3}, (2, - 4)

\int \frac{cos ( 3 x )}{sin ( 3 x )} d x

\int \frac{cos ( 2 x )}{cos ^{2} ( x ) sin ^{2} ( x )} d x

What is the integral from 0 to 5 of x^2(x-5)^4 ?
The integral from 0 to 5 of x^2(x-5)^4 is 15625/21