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

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

integral of x(x-2)^5

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

\frac{( x - 2 ) ^{7}}{7} + \frac{( x - 2 ) ^{6}}{3} + C

Solution steps

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

Apply u-substitution

= \int (u + 2) u^{5} d u

Expand

(u + 2) u^{5} : u^{6} + 2 u^{5}

= \int u^{6} + 2 u^{5} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{6} d u + \int 2 u^{5} d u

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

\int 2 u^{5} d u = \frac{u ^{6}}{3}

= \frac{u ^{7}}{7} + \frac{u ^{6}}{3}

Substitute back

u = x - 2

= \frac{( x - 2 ) ^{7}}{7} + \frac{( x - 2 ) ^{6}}{3}

Add a constant to the solution

= \frac{( x - 2 ) ^{7}}{7} + \frac{( x - 2 ) ^{6}}{3} + C

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