Question 1025369
{{{int(x*sqrt(1-x),dx)}}}
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Integrate by parts,
{{{int(v,du)=uv-int(u,dv)}}}
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{{{du=(1-x)^(1/2)dx}}}
{{{v=x}}}
So then,
{{{u=int((1-x)^(1/2),dx)=-(2/3)(1-x)^(3/2)}}}
{{{dv=dx}}}
and 
{{{uv=-(2/3)x(1-x)^(3/2)}}}
{{{int(u,dv)=-(2/3)int((1-x)^(3/2),dx)=-(2/3)(-(2/5))(1-x)^(5/2)=(4/15)(1-x)^(5/2)}}}
Putting it all together,
{{{int(x*sqrt(1-x),dx)=-(2/3)x(1-x)^(3/2)-(4/15)(1-x)^(5/2)}}}
{{{int(x*sqrt(1-x),dx)=(1-x)^(3/2)(-(10/15)x-(4/15)(1-x))}}}
{{{int(x*sqrt(1-x),dx)=(1/15)(1-x)^(3/2)(-10x-4(1-x))}}}
{{{int(x*sqrt(1-x),dx)=(1/15)(1-x)^(3/2)(-10x-4+4x)}}}
{{{int(x*sqrt(1-x),dx)=(1/15)(1-x)^(3/2)(-6x-4)}}}
{{{int(x*sqrt(1-x),dx)=-(2/15)(1-x)^(3/2)(3x+2)}}}