Mister Exam

Integral of x(e^(2x)) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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01xe2xdx\int\limits_{0}^{1} x e^{2 x}\, dx
Integral(x*E^(2*x), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=xu{\left(x \right)} = x and let dv(x)=e2x\operatorname{dv}{\left(x \right)} = e^{2 x}.

    Then du(x)=1\operatorname{du}{\left(x \right)} = 1.

    To find v(x)v{\left(x \right)}:

    1. There are multiple ways to do this integral.

      Method #1

      1. Let u=2xu = 2 x.

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

        eu4du\int \frac{e^{u}}{4}\, du

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

          eu2du=eudu2\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}

          1. The integral of the exponential function is itself.

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

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

        Now substitute uu back in:

        e2x2\frac{e^{2 x}}{2}

      Method #2

      1. Let u=e2xu = e^{2 x}.

        Then let du=2e2xdxdu = 2 e^{2 x} dx and substitute du2\frac{du}{2}:

        14du\int \frac{1}{4}\, du

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

          12du=1du2\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}

          1. The integral of a constant is the constant times the variable of integration:

            1du=u\int 1\, du = u

          So, the result is: u2\frac{u}{2}

        Now substitute uu back in:

        e2x2\frac{e^{2 x}}{2}

    Now evaluate the sub-integral.

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

    e2x2dx=e2xdx2\int \frac{e^{2 x}}{2}\, dx = \frac{\int e^{2 x}\, dx}{2}

    1. Let u=2xu = 2 x.

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

      eu4du\int \frac{e^{u}}{4}\, du

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

        eu2du=eudu2\int \frac{e^{u}}{2}\, du = \frac{\int e^{u}\, du}{2}

        1. The integral of the exponential function is itself.

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

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

      Now substitute uu back in:

      e2x2\frac{e^{2 x}}{2}

    So, the result is: e2x4\frac{e^{2 x}}{4}

  3. Now simplify:

    (2x1)e2x4\frac{\left(2 x - 1\right) e^{2 x}}{4}

  4. Add the constant of integration:

    (2x1)e2x4+constant\frac{\left(2 x - 1\right) e^{2 x}}{4}+ \mathrm{constant}


The answer is:

(2x1)e2x4+constant\frac{\left(2 x - 1\right) e^{2 x}}{4}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                             
 |                  2*x      2*x
 |    2*x          e      x*e   
 | x*e    dx = C - ---- + ------
 |                  4       2   
/                               
(2x1)e2x4{{\left(2\,x-1\right)\,e^{2\,x}}\over{4}}
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1010
The answer [src]
     2
1   e 
- + --
4   4 
e24+14{{e^2}\over{4}}+{{1}\over{4}}
=
=
     2
1   e 
- + --
4   4 
14+e24\frac{1}{4} + \frac{e^{2}}{4}
Numerical answer [src]
2.09726402473266
2.09726402473266
The graph
Integral of x(e^(2x)) dx

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