What is the integral of (3x^2-5x)^3(6x-5) ?

The integral of (3x^2-5x)^3(6x-5) is (81x^8)/4-135x^7+(675x^6)/2-375x^5+(625x^4)/4+C

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integral of (3x^2-5x)^3(6x-5)

\int (3 x^{2} - 5 x)^{3} (6 x - 5) d x

\frac{81 x ^{8}}{4} - 135 x^{7} + \frac{675 x ^{6}}{2} - 375 x^{5} + \frac{625 x ^{4}}{4} + C

Solution steps

\int (3 x^{2} - 5 x)^{3} (6 x - 5) d x

Expand

(3 x^{2} - 5 x)^{3} (6 x - 5) : 162 x^{7} - 945 x^{6} + 2025 x^{5} - 1875 x^{4} + 625 x^{3}

= \int 162 x^{7} - 945 x^{6} + 2025 x^{5} - 1875 x^{4} + 625 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

= \int 162 x^{7} d x - \int 945 x^{6} d x + \int 2025 x^{5} d x - \int 1875 x^{4} d x + \int 625 x^{3} d x

\int 162 x^{7} d x = \frac{81 x ^{8}}{4}

\int 945 x^{6} d x = 135 x^{7}

\int 2025 x^{5} d x = \frac{675 x ^{6}}{2}

\int 1875 x^{4} d x = 375 x^{5}

\int 625 x^{3} d x = \frac{625 x ^{4}}{4}

= \frac{81 x ^{8}}{4} - 135 x^{7} + \frac{675 x ^{6}}{2} - 375 x^{5} + \frac{625 x ^{4}}{4}

Add a constant to the solution

= \frac{81 x ^{8}}{4} - 135 x^{7} + \frac{675 x ^{6}}{2} - 375 x^{5} + \frac{625 x ^{4}}{4} + C

y (9 - y) - \frac{81}{4} = \frac{d y}{d t}, y (0) = 4

x \to \infty lim (1. 2^{x})

x \to 0 lim (\frac{2}{1 + e ^{\frac{1}{x}}})

t \frac{d y}{d t} = t^{3} + 14 t^{3} y

\frac{d}{d x} (- \frac{y}{x ^{2} + y ^{2} + 1})

What is the integral of (3x^2-5x)^3(6x-5) ?
The integral of (3x^2-5x)^3(6x-5) is (81x^8)/4-135x^7+(675x^6)/2-375x^5+(625x^4)/4+C