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

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

Limits of integration:

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The solution

You have entered [src] 
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∫01ex2−1x dx\int\limits_{0}^{1} e^{x^{2} - 1} x\, dx 
Integral(x*E^(x^2 - 1), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=x2−1u = x^{2} - 1.

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

      ∫eu2 du\int \frac{e^{u}}{2}\, du

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

        False\text{False}

        1. The integral of the exponential function is itself.

          ∫eu du=eu\int e^{u}\, du = e^{u}

        So, the result is: eu2\frac{e^{u}}{2}

      Now substitute uu back in:

      ex2−12\frac{e^{x^{2} - 1}}{2}

    Method #2

    1. Rewrite the integrand:

      ex2−1x=xex2ee^{x^{2} - 1} x = \frac{x e^{x^{2}}}{e}

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

      ∫xex2e dx=∫xex2 dxe\int \frac{x e^{x^{2}}}{e}\, dx = \frac{\int x e^{x^{2}}\, dx}{e}

      1. Let u=x2u = x^{2}.

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

        ∫eu2 du\int \frac{e^{u}}{2}\, du

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

          False\text{False}

          1. The integral of the exponential function is itself.

            ∫eu du=eu\int e^{u}\, du = e^{u}

          So, the result is: eu2\frac{e^{u}}{2}

        Now substitute uu back in:

        ex22\frac{e^{x^{2}}}{2}

      So, the result is: ex22e\frac{e^{x^{2}}}{2 e}

    Method #3

    1. Rewrite the integrand:

      ex2−1x=xex2ee^{x^{2} - 1} x = \frac{x e^{x^{2}}}{e}

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

      ∫xex2e dx=∫xex2 dxe\int \frac{x e^{x^{2}}}{e}\, dx = \frac{\int x e^{x^{2}}\, dx}{e}

      1. Let u=x2u = x^{2}.

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

        ∫eu2 du\int \frac{e^{u}}{2}\, du

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

          False\text{False}

          1. The integral of the exponential function is itself.

            ∫eu du=eu\int e^{u}\, du = e^{u}

          So, the result is: eu2\frac{e^{u}}{2}

        Now substitute uu back in:

        ex22\frac{e^{x^{2}}}{2}

      So, the result is: ex22e\frac{e^{x^{2}}}{2 e}

  2. Now simplify:

    ex2−12\frac{e^{x^{2} - 1}}{2}

  3. Add the constant of integration:

    ex2−12+constant\frac{e^{x^{2} - 1}}{2}+ \mathrm{constant}

The answer is:

ex2−12+constant\frac{e^{x^{2} - 1}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                          
 |                      2    
 |     2               x  - 1
 |    x  - 1          e      
 | x*E       dx = C + -------
 |                       2   
/
∫ex2−1x dx=C+ex2−12\int e^{x^{2} - 1} x\, dx = C + \frac{e^{x^{2} - 1}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
Plot the graph f(x)
The answer [src] 
     -1
1   e  
- - ---
2    2
12−12e\frac{1}{2} - \frac{1}{2 e} 
=
= 
     -1
1   e  
- - ---
2    2
12−12e\frac{1}{2} - \frac{1}{2 e} 
1/2 - exp(-1)/2
Numerical answer [src] 
0.316060279414279
0.316060279414279

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

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