anonymous
  • anonymous
Integrate (x) (1-x)^(1/2) dx not sure where to even start....thanks
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
try u = 1-x
anonymous
  • anonymous
making x = 1-u
anonymous
  • anonymous
\[\int\limits\limits\limits_{?}^{?} x \sqrt{1-x} dx\]

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anonymous
  • anonymous
giving \[-\int(1-u)\sqrt(u)du=-\int u^{\frac{1}{2}}-u^{\frac{3}{2}}du\] etc
anonymous
  • anonymous
using equation thing is not very intuitive is it? trying u = 1-x
anonymous
  • anonymous
well no, but it is a standard trick.
anonymous
  • anonymous
you have to deal with the annoying 1-x inside the radical somehow.
anonymous
  • anonymous
so even though the derivative of 1-x would be -1 I just keep marching?
anonymous
  • anonymous
yes. you have \[u=1-x\] \[du=-dx\] or \[dx=-du\] and since \[u=1-x\] we know \[x=1-u\] to get \[\int x sqrt{1-x}dx=-\int (1-u) \sqrt{u}du = \int u^{\frac{3}{2}}du - \int u^{\frac{1}{2}}du\]
anonymous
  • anonymous
i assume it is ok from there. take the antiderivative using the power rule and then replace u by 1-x
anonymous
  • anonymous
- 2/3 (1-x)^ 3/2 + 2/5 (1-x)^5/2 +C
anonymous
  • anonymous
that is it!

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