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x*exp(4*x)
Solving integrals/ x*exp(4*x)

Integral of x*exp(4*x) dx

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The solution

You have entered [src] 
  1          
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 |     4*x   
 |  x*e    dx
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0
∫01xe4x dx\int\limits_{0}^{1} x e^{4 x}\, dx 
Integral(x*exp(4*x), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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)=e4x\operatorname{dv}{\left(x \right)} = e^{4 x}.

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

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

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫eu4 du\int \frac{e^{u}}{4}\, 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: eu4\frac{e^{u}}{4}

      Now substitute uu back in:

      e4x4\frac{e^{4 x}}{4}

    Now evaluate the sub-integral.

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

    ∫e4x4 dx=∫e4x dx4\int \frac{e^{4 x}}{4}\, dx = \frac{\int e^{4 x}\, dx}{4}

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫eu4 du\int \frac{e^{u}}{4}\, 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: eu4\frac{e^{u}}{4}

      Now substitute uu back in:

      e4x4\frac{e^{4 x}}{4}

    So, the result is: e4x16\frac{e^{4 x}}{16}

  3. Now simplify:

    (4x−1)e4x16\frac{\left(4 x - 1\right) e^{4 x}}{16}

  4. Add the constant of integration:

    (4x−1)e4x16+constant\frac{\left(4 x - 1\right) e^{4 x}}{16}+ \mathrm{constant}

The answer is:

(4x−1)e4x16+constant\frac{\left(4 x - 1\right) e^{4 x}}{16}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                             
 |                  4*x      4*x
 |    4*x          e      x*e   
 | x*e    dx = C - ---- + ------
 |                  16      4   
/
∫xe4x dx=C+xe4x4−e4x16\int x e^{4 x}\, dx = C + \frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-50100
Plot the graph f(x)
The answer [src] 
        4
1    3*e 
-- + ----
16    16
116+3e416\frac{1}{16} + \frac{3 e^{4}}{16} 
=
= 
        4
1    3*e 
-- + ----
16    16
116+3e416\frac{1}{16} + \frac{3 e^{4}}{16} 
1/16 + 3*exp(4)/16
Numerical answer [src] 
10.2996531312145
10.2996531312145
The graph
Integral of x*exp(4*x) dx

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

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