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

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

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The solution

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 |      2       
 |     x  + 1   
 |  x*E       dx
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0
∫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=exex2e^{x^{2} + 1} x = e x e^{x^{2}}

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

      ∫exex2 dx=e∫xex2 dx\int e x e^{x^{2}}\, dx = e \int x e^{x^{2}}\, dx

      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: eex22\frac{e e^{x^{2}}}{2}

    Method #3

    1. Rewrite the integrand:

      ex2+1x=exex2e^{x^{2} + 1} x = e x e^{x^{2}}

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

      ∫exex2 dx=e∫xex2 dx\int e x e^{x^{2}}\, dx = e \int x e^{x^{2}}\, dx

      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: eex22\frac{e e^{x^{2}}}{2}

  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
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
 2    
e    E
-- - -
2    2
−e2+e22- \frac{e}{2} + \frac{e^{2}}{2} 
=
= 
 2    
e    E
-- - -
2    2
−e2+e22- \frac{e}{2} + \frac{e^{2}}{2} 
exp(2)/2 - E/2
Numerical answer [src] 
2.3353871352358
2.3353871352358
The graph
Integral of x*e^(x^2+1) dx

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

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