What is the integral of x(1-x)^5 ?

The integral of x(1-x)^5 is 1/2 x^2(1-x)^5+1/42 (35x^3-105x^4+126x^5-70x^6+15x^7)+C

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integral of x(1-x)^5

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

\frac{1}{2} x^{2} (1 - x)^{5} + \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7}) + C

Solution steps

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

Apply Integration By Parts

= \frac{1}{2} x^{2} (1 - x)^{5} - \int - \frac{5}{2} x^{2} (1 - x)^{4} d x

\int - \frac{5}{2} x^{2} (1 - x)^{4} d x = - \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7})

= \frac{1}{2} x^{2} (1 - x)^{5} - (- \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7}))

Simplify

= \frac{1}{2} x^{2} (1 - x)^{5} + \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7})

Add a constant to the solution

= \frac{1}{2} x^{2} (1 - x)^{5} + \frac{1}{42} (35 x^{3} - 105 x^{4} + 126 x^{5} - 70 x^{6} + 15 x^{7}) + C

What is the integral of x(1-x)^5 ?
The integral of x(1-x)^5 is 1/2 x^2(1-x)^5+1/42 (35x^3-105x^4+126x^5-70x^6+15x^7)+C