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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

You have entered [src] 
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∫01xx2+1 dx\int\limits_{0}^{1} x \sqrt{x^{2} + 1}\, dx 
Integral(x*sqrt(1 + x^2), (x, 0, 1))
Detail solution

  1. Let u=x2+1u = x^{2} + 1.

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

    ∫u2 du\int \frac{\sqrt{u}}{2}\, 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:

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

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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 | x*\/  1 + x   dx = C + -----------
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∫xx2+1 dx=C+(x2+1)323\int x \sqrt{x^{2} + 1}\, dx = C + \frac{\left(x^{2} + 1\right)^{\frac{3}{2}}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
          ___
  1   2*\/ 2 
- - + -------
  3      3
−13+223- \frac{1}{3} + \frac{2 \sqrt{2}}{3} 
=
= 
          ___
  1   2*\/ 2 
- - + -------
  3      3
−13+223- \frac{1}{3} + \frac{2 \sqrt{2}}{3} 
-1/3 + 2*sqrt(2)/3
Numerical answer [src] 
0.60947570824873
0.60947570824873
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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