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

The integral of x(x-3)^5 is (x^7)/7-(5x^6)/2+18x^5-(135x^4)/2+135x^3-(243x^2)/2+C

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integral of x(x-3)^5

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

\frac{x ^{7}}{7} - \frac{5 x ^{6}}{2} + 18 x^{5} - \frac{135 x ^{4}}{2} + 135 x^{3} - \frac{243 x ^{2}}{2} + C

Solution steps

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

Expand

x (x - 3)^{5} : x^{6} - 15 x^{5} + 90 x^{4} - 270 x^{3} + 405 x^{2} - 243 x

= \int x^{6} - 15 x^{5} + 90 x^{4} - 270 x^{3} + 405 x^{2} - 243 x 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^{6} d x - \int 15 x^{5} d x + \int 90 x^{4} d x - \int 270 x^{3} d x + \int 405 x^{2} d x - \int 243 x d x

\int x^{6} d x = \frac{x ^{7}}{7}

\int 15 x^{5} d x = \frac{5 x ^{6}}{2}

\int 90 x^{4} d x = 18 x^{5}

\int 270 x^{3} d x = \frac{135 x ^{4}}{2}

\int 405 x^{2} d x = 135 x^{3}

\int 243 x d x = \frac{243 x ^{2}}{2}

= \frac{x ^{7}}{7} - \frac{5 x ^{6}}{2} + 18 x^{5} - \frac{135 x ^{4}}{2} + 135 x^{3} - \frac{243 x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{7}}{7} - \frac{5 x ^{6}}{2} + 18 x^{5} - \frac{135 x ^{4}}{2} + 135 x^{3} - \frac{243 x ^{2}}{2} + C

y = 1 + 3 x, 3 \leq x \leq 9

y = 10 x^{2}, y = x^{2} + 12

\int e^{\frac{- 4 x}{3}} d x

x \to 0 lim (e^{x} - 1 - x)

n = 0 \sum \infty (\frac{2}{5})^{n}

What is the integral of x(x-3)^5 ?
The integral of x(x-3)^5 is (x^7)/7-(5x^6)/2+18x^5-(135x^4)/2+135x^3-(243x^2)/2+C