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e^(2x+3)
Solving integrals/ e^(2x+3)

Integral of e^(2x+3) dx

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The solution

You have entered [src] 
  1            
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 |   2*x + 3   
 |  E        dx
 |             
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0
∫01e2x+3 dx\int\limits_{0}^{1} e^{2 x + 3}\, dx 
Integral(E^(2*x + 3), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=2x+3u = 2 x + 3.

      Then let du=2dxdu = 2 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:

      e2x+32\frac{e^{2 x + 3}}{2}

    Method #2

    1. Rewrite the integrand:

      e2x+3=e3e2xe^{2 x + 3} = e^{3} e^{2 x}

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

      ∫e3e2x dx=e3∫e2x dx\int e^{3} e^{2 x}\, dx = e^{3} \int e^{2 x}\, dx

      1. Let u=2xu = 2 x.

        Then let du=2dxdu = 2 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:

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

      So, the result is: e3e2x2\frac{e^{3} e^{2 x}}{2}

    Method #3

    1. Rewrite the integrand:

      e2x+3=e3e2xe^{2 x + 3} = e^{3} e^{2 x}

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

      ∫e3e2x dx=e3∫e2x dx\int e^{3} e^{2 x}\, dx = e^{3} \int e^{2 x}\, dx

      1. Let u=2xu = 2 x.

        Then let du=2dxdu = 2 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:

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

      So, the result is: e3e2x2\frac{e^{3} e^{2 x}}{2}

  2. Now simplify:

    e2x+32\frac{e^{2 x + 3}}{2}

  3. Add the constant of integration:

    e2x+32+constant\frac{e^{2 x + 3}}{2}+ \mathrm{constant}

The answer is:

e2x+32+constant\frac{e^{2 x + 3}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                          
 |                    2*x + 3
 |  2*x + 3          e       
 | E        dx = C + --------
 |                      2    
/
∫e2x+3 dx=C+e2x+32\int e^{2 x + 3}\, dx = C + \frac{e^{2 x + 3}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900200
Plot the graph f(x)
The answer [src] 
 5    3
e    e 
-- - --
2    2
−e32+e52- \frac{e^{3}}{2} + \frac{e^{5}}{2} 
=
= 
 5    3
e    e 
-- - --
2    2
−e32+e52- \frac{e^{3}}{2} + \frac{e^{5}}{2} 
exp(5)/2 - exp(3)/2
Numerical answer [src] 
64.1638110896945
64.1638110896945
The graph
Integral of e^(2x+3) dx

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

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