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e^(5x+7)*dx
Solving integrals/ e^(5x+7)*dx

Integral of e^(5x+7)*dx dx

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The solution

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  1              
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 |   5*x + 7     
 |  e       *1 dx
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0
∫01e5x+7⋅1 dx\int\limits_{0}^{1} e^{5 x + 7} \cdot 1\, dx 
Integral(E^(5*x + 7)*1, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      e5x+7⋅1=e7e5xe^{5 x + 7} \cdot 1 = e^{7} e^{5 x}

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

      ∫e7e5x dx=e7∫e5x dx\int e^{7} e^{5 x}\, dx = e^{7} \int e^{5 x}\, dx

      1. There are multiple ways to do this integral.

        Method #1

        1. Let u=5xu = 5 x.

          Then let du=5dxdu = 5 dx and substitute du5\frac{du}{5}:

          ∫eu25 du\int \frac{e^{u}}{25}\, du

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

            ∫eu5 du=∫eu du5\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}

            1. The integral of the exponential function is itself.

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

            So, the result is: eu5\frac{e^{u}}{5}

          Now substitute uu back in:

          e5x5\frac{e^{5 x}}{5}

        Method #2

        1. Let u=e5xu = e^{5 x}.

          Then let du=5e5xdxdu = 5 e^{5 x} dx and substitute du5\frac{du}{5}:

          ∫125 du\int \frac{1}{25}\, du

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

            ∫15 du=∫1 du5\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u5\frac{u}{5}

          Now substitute uu back in:

          e5x5\frac{e^{5 x}}{5}

      So, the result is: e7e5x5\frac{e^{7} e^{5 x}}{5}

    Method #2

    1. Rewrite the integrand:

      e5x+7⋅1=e7e5xe^{5 x + 7} \cdot 1 = e^{7} e^{5 x}

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

      ∫e7e5x dx=e7∫e5x dx\int e^{7} e^{5 x}\, dx = e^{7} \int e^{5 x}\, dx

      1. Let u=5xu = 5 x.

        Then let du=5dxdu = 5 dx and substitute du5\frac{du}{5}:

        ∫eu25 du\int \frac{e^{u}}{25}\, du

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

          ∫eu5 du=∫eu du5\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}

          1. The integral of the exponential function is itself.

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

          So, the result is: eu5\frac{e^{u}}{5}

        Now substitute uu back in:

        e5x5\frac{e^{5 x}}{5}

      So, the result is: e7e5x5\frac{e^{7} e^{5 x}}{5}

  2. Now simplify:

    e5x+75\frac{e^{5 x + 7}}{5}

  3. Add the constant of integration:

    e5x+75+constant\frac{e^{5 x + 7}}{5}+ \mathrm{constant}

The answer is:

e5x+75+constant\frac{e^{5 x + 7}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                           
 |                      7  5*x
 |  5*x + 7            e *e   
 | e       *1 dx = C + -------
 |                        5   
/
e5 x+75{{e^{5\,x+7}}\over{5}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900200000
Plot the graph f(x)
The answer [src] 
   7    12
  e    e  
- -- + ---
  5     5
e125−e75{{e^{12}}\over{5}}-{{e^7}\over{5}} 
=
= 
   7    12
  e    e  
- -- + ---
  5     5
−e75+e125- \frac{e^{7}}{5} + \frac{e^{12}}{5}
Simplify
Numerical answer [src] 
32331.6316521151
32331.6316521151
The graph
Integral of e^(5x+7)*dx dx

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

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