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

The integral of (x-3)(x^2-6x+2)^3 is (x^8)/8-3x^7+28x^6-126x^5+273x^4-252x^3+112x^2-24x+C

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

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

\frac{x ^{8}}{8} - 3 x^{7} + 28 x^{6} - 126 x^{5} + 273 x^{4} - 252 x^{3} + 112 x^{2} - 24 x + C

Solution steps

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

Expand

(x - 3) (x^{2} - 6 x + 2)^{3} : x^{7} - 21 x^{6} + 168 x^{5} - 630 x^{4} + 1092 x^{3} - 756 x^{2} + 224 x - 24

= \int x^{7} - 21 x^{6} + 168 x^{5} - 630 x^{4} + 1092 x^{3} - 756 x^{2} + 224 x - 24 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 x^{7} d x - \int 21 x^{6} d x + \int 168 x^{5} d x - \int 630 x^{4} d x + \int 1092 x^{3} d x - \int 756 x^{2} d x + \int 224 x d x - \int 24 d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int 21 x^{6} d x = 3 x^{7}

\int 168 x^{5} d x = 28 x^{6}

\int 630 x^{4} d x = 126 x^{5}

\int 1092 x^{3} d x = 273 x^{4}

\int 756 x^{2} d x = 252 x^{3}

\int 224 x d x = 112 x^{2}

\int 24 d x = 24 x

= \frac{x ^{8}}{8} - 3 x^{7} + 28 x^{6} - 126 x^{5} + 273 x^{4} - 252 x^{3} + 112 x^{2} - 24 x

Add a constant to the solution

= \frac{x ^{8}}{8} - 3 x^{7} + 28 x^{6} - 126 x^{5} + 273 x^{4} - 252 x^{3} + 112 x^{2} - 24 x + C

\int_{0}^{6} e^{x} sin (x) d x

y = 2 x, (1, 3)

\int sin (18 x) cos (13 x) d x

\int x^{2} \cdot x d x

(t - 1)^{2}

What is the integral of (x-3)(x^2-6x+2)^3 ?
The integral of (x-3)(x^2-6x+2)^3 is (x^8)/8-3x^7+28x^6-126x^5+273x^4-252x^3+112x^2-24x+C