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1/sqrt(x)*sqrt(1-x)
Solving integrals/ 1/sqrt(x)*sqrt(1-x)

Integral of 1/sqrt(x)*sqrt(1-x) dx

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The solution

You have entered [src] 
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∫011⋅1x−x+1 dx\int\limits_{0}^{1} 1 \cdot \frac{1}{\sqrt{x}} \sqrt{- x + 1}\, dx 
Integral(1*sqrt(1 - x)/sqrt(x), (x, 0, 1))
Detail solution

  1. Let u=xu = \sqrt{x}.

    Then let du=dx2xdu = \frac{dx}{2 \sqrt{x}} and substitute 2du2 du:

    ∫41−u2 du\int 4 \sqrt{1 - u^{2}}\, du

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫21−u2 du=2∫1−u2 du\int 2 \sqrt{1 - u^{2}}\, du = 2 \int \sqrt{1 - u^{2}}\, du

        SqrtQuadraticRule(a=1, b=0, c=-1, context=sqrt(1 - _u**2), symbol=_u)

      So, the result is: u1−u2+asin⁡(u)u \sqrt{1 - u^{2}} + \operatorname{asin}{\left(u \right)}

    Now substitute uu back in:

    x1−x+asin⁡(x)\sqrt{x} \sqrt{1 - x} + \operatorname{asin}{\left(\sqrt{x} \right)}

  2. Add the constant of integration:

    x1−x+asin⁡(x)+constant\sqrt{x} \sqrt{1 - x} + \operatorname{asin}{\left(\sqrt{x} \right)}+ \mathrm{constant}

The answer is:

x1−x+asin⁡(x)+constant\sqrt{x} \sqrt{1 - x} + \operatorname{asin}{\left(\sqrt{x} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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 | 1*-----*\/ 1 - x  dx = C + \/ x *\/ 1 - x  + asin\\/ x /
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1−x(1−xx+1) x−arctan⁡(1−xx){{\sqrt{1-x}}\over{\left({{1-x}\over{x}}+1\right)\,\sqrt{x}}}- \arctan \left({{\sqrt{1-x}}\over{\sqrt{x}}}\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900200
Plot the graph f(x)
The answer [src] 
pi
--
2
π2{{\pi}\over{2}} 
=
= 
pi
--
2
π2\frac{\pi}{2}
Simplify
Numerical answer [src] 
1.57079632626431
1.57079632626431
The graph
Integral of 1/sqrt(x)*sqrt(1-x) dx

    Use the examples entering the upper and lower limits of integration.

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