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x*sqrt(1-x^2)
Solving integrals/ x*sqrt(1-x^2)

Integral of x*sqrt(1-x^2) dx

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The solution

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∫01x1−x2 dx\int\limits_{0}^{1} x \sqrt{1 - x^{2}}\, dx 
Integral(x*sqrt(1 - x^2), (x, 0, 1))
Detail solution

  1. Let u=1−x2u = 1 - x^{2}.

    Then let du=−2xdxdu = - 2 x dx and substitute −du2- \frac{du}{2}:

    ∫(−u2) du\int \left(- \frac{\sqrt{u}}{2}\right)\, du

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

      ∫u du=−∫u du2\int \sqrt{u}\, du = - \frac{\int \sqrt{u}\, du}{2}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫u du=2u323\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}

      So, the result is: −u323- \frac{u^{\frac{3}{2}}}{3}

    Now substitute uu back in:

    −(1−x2)323- \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{3}

  2. Add the constant of integration:

    −(1−x2)323+constant- \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer is:

−(1−x2)323+constant- \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫x1−x2 dx=C−(1−x2)323\int x \sqrt{1 - x^{2}}\, dx = C - \frac{\left(1 - x^{2}\right)^{\frac{3}{2}}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
1/3
13\frac{1}{3} 
=
= 
1/3
13\frac{1}{3} 
1/3
Numerical answer [src] 
0.333333333333333
0.333333333333333
The graph
Integral of x*sqrt(1-x^2) dx

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

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