Answer:
[tex]-\frac{2}{15}(1-x)^\frac{3}{2}(3x+2)+C[/tex]
Step-by-step explanation:
Use the method of integration by parts. The method is based in the following rule of derivation of two functions f and g :
[tex](fg)' = f'g+fg'\implies fg'=(fg)'-f'g\implies \\\int fg' dx = fg -\int f'g dx[/tex]
Identify suitable f and g in the original expression. I use:
[tex]f=x\,\,\,, \,\,\,g=\sqrt{1-x}\\f'=1\,\,\,,\,\,\,g'=-\frac{2}{3}(1-x)^\frac{3}{2}[/tex]
and write the integration by parts :
[tex]\int x\sqrt{1-x} dx = -\frac{2}{3}x(1-x)^\frac{3}{2}+\frac{2}{3}\int (1-x)^\frac{3}{2}dx=\\ =-\frac{2}{3}x(1-x)^\frac{3}{2}-\frac{4}{15}(1-x)^\frac{5}{2}+C=\\=-\frac{2}{15}(1-x)^\frac{3}{2}(3x+2)+C[/tex]
with C an arbitrary integration constant.
