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

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

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The solution

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 |   5*x - 7   
 |  E        dx
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∫00e5x−7 dx\int\limits_{0}^{0} e^{5 x - 7}\, dx 
Integral(E^(5*x - 7), (x, 0, 0))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=5x−7u = 5 x - 7.

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

      ∫eu5 du\int \frac{e^{u}}{5}\, 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: eu5\frac{e^{u}}{5}

      Now substitute uu back in:

      e5x−75\frac{e^{5 x - 7}}{5}

    Method #2

    1. Rewrite the integrand:

      e5x−7=e5xe7e^{5 x - 7} = \frac{e^{5 x}}{e^{7}}

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

      ∫e5xe7 dx=∫e5x dxe7\int \frac{e^{5 x}}{e^{7}}\, dx = \frac{\int e^{5 x}\, dx}{e^{7}}

      1. Let u=5xu = 5 x.

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

        ∫eu5 du\int \frac{e^{u}}{5}\, 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: eu5\frac{e^{u}}{5}

        Now substitute uu back in:

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

      So, the result is: e5x5e7\frac{e^{5 x}}{5 e^{7}}

    Method #3

    1. Rewrite the integrand:

      e5x−7=e5xe7e^{5 x - 7} = \frac{e^{5 x}}{e^{7}}

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

      ∫e5xe7 dx=∫e5x dxe7\int \frac{e^{5 x}}{e^{7}}\, dx = \frac{\int e^{5 x}\, dx}{e^{7}}

      1. Let u=5xu = 5 x.

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

        ∫eu5 du\int \frac{e^{u}}{5}\, 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: eu5\frac{e^{u}}{5}

        Now substitute uu back in:

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

      So, the result is: e5x5e7\frac{e^{5 x}}{5 e^{7}}

  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] 
  /                          
 |                    5*x - 7
 |  5*x - 7          e       
 | E        dx = C + --------
 |                      5    
/
∫e5x−7 dx=C+e5x−75\int e^{5 x - 7}\, dx = C + \frac{e^{5 x - 7}}{5}
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    Use the examples entering the upper and lower limits of integration.

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