What is the integral of (x-1)^6 ?

The integral of (x-1)^6 is x(x-1)^6-6/7 (x-1)^7-(x-1)^6+C

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integral of (x-1)^6

\int (x - 1)^{6} d x

x (x - 1)^{6} - \frac{6}{7} (x - 1)^{7} - (x - 1)^{6} + C

Solution steps

\int (x - 1)^{6} d x

Apply Integration By Parts

: (x - 1)^{6} x - \int 6 (x - 1)^{5} x d x

= (x - 1)^{6} x - \int 6 (x - 1)^{5} x d x

\int 6 (x - 1)^{5} x d x = \frac{6}{7} (x - 1)^{7} + (x - 1)^{6}

= (x - 1)^{6} x - (\frac{6}{7} (x - 1)^{7} + (x - 1)^{6})

Simplify

(x - 1)^{6} x - (\frac{6}{7} (x - 1)^{7} + (x - 1)^{6}) : x (x - 1)^{6} - \frac{6}{7} (x - 1)^{7} - (x - 1)^{6}

= x (x - 1)^{6} - \frac{6}{7} (x - 1)^{7} - (x - 1)^{6}

Add a constant to the solution

= x (x - 1)^{6} - \frac{6}{7} (x - 1)^{7} - (x - 1)^{6} + C

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What is the integral of (x-1)^6 ?
The integral of (x-1)^6 is x(x-1)^6-6/7 (x-1)^7-(x-1)^6+C