What is the integral of x(x-1)^7 ?

The integral of x(x-1)^7 is (x^9)/9-(7x^8)/8+3x^7-(35x^6)/6+7x^5-(21x^4)/4+(7x^3)/3-(x^2)/2+C

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integral of x(x-1)^7

\int x (x - 1)^{7} d x

\frac{x ^{9}}{9} - \frac{7 x ^{8}}{8} + 3 x^{7} - \frac{35 x ^{6}}{6} + 7 x^{5} - \frac{21 x ^{4}}{4} + \frac{7 x ^{3}}{3} - \frac{x ^{2}}{2} + C

Solution steps

\int x (x - 1)^{7} d x

Expand

x (x - 1)^{7} : x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - x

= \int x^{8} - 7 x^{7} + 21 x^{6} - 35 x^{5} + 35 x^{4} - 21 x^{3} + 7 x^{2} - 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^{8} d x - \int 7 x^{7} d x + \int 21 x^{6} d x - \int 35 x^{5} d x + \int 35 x^{4} d x - \int 21 x^{3} d x + \int 7 x^{2} d x - \int x d x

\int x^{8} d x = \frac{x ^{9}}{9}

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

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

\int 35 x^{5} d x = \frac{35 x ^{6}}{6}

\int 35 x^{4} d x = 7 x^{5}

\int 21 x^{3} d x = \frac{21 x ^{4}}{4}

\int 7 x^{2} d x = \frac{7 x ^{3}}{3}

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

= \frac{x ^{9}}{9} - \frac{7 x ^{8}}{8} + 3 x^{7} - \frac{35 x ^{6}}{6} + 7 x^{5} - \frac{21 x ^{4}}{4} + \frac{7 x ^{3}}{3} - \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{9}}{9} - \frac{7 x ^{8}}{8} + 3 x^{7} - \frac{35 x ^{6}}{6} + 7 x^{5} - \frac{21 x ^{4}}{4} + \frac{7 x ^{3}}{3} - \frac{x ^{2}}{2} + C

\int e + sec^{2} (θ) - e^{θ} d θ

cot^{3} (4 x + 1)

\frac{d}{d x} (- x^{4} + x)

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

\int (3 - \frac{1}{x})^{- 2} (\frac{1}{x ^{2}}) d x

What is the integral of x(x-1)^7 ?
The integral of x(x-1)^7 is (x^9)/9-(7x^8)/8+3x^7-(35x^6)/6+7x^5-(21x^4)/4+(7x^3)/3-(x^2)/2+C