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

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 5 x - 7$.

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

      $$\int \frac{e^{u}}{5}\, du$$

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

         $$\text{False}$$

         1. The integral of the exponential function is itself.

            $$\int e^{u}\, du = e^{u}$$

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

      Now substitute $u$ back in:

      $$\frac{e^{5 x - 7}}{5}$$

   Method #2

   1. Rewrite the integrand:

      $$e^{5 x - 7} = \frac{e^{5 x}}{e^{7}}$$

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

      $$\int \frac{e^{5 x}}{e^{7}}\, dx = \frac{\int e^{5 x}\, dx}{e^{7}}$$

      1. Let $u = 5 x$.

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

         $$\int \frac{e^{u}}{5}\, du$$

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

            $$\text{False}$$

            1. The integral of the exponential function is itself.

               $$\int e^{u}\, du = e^{u}$$

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

         Now substitute $u$ back in:

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

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

   Method #3

   1. Rewrite the integrand:

      $$e^{5 x - 7} = \frac{e^{5 x}}{e^{7}}$$

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

      $$\int \frac{e^{5 x}}{e^{7}}\, dx = \frac{\int e^{5 x}\, dx}{e^{7}}$$

      1. Let $u = 5 x$.

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

         $$\int \frac{e^{u}}{5}\, du$$

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

            $$\text{False}$$

            1. The integral of the exponential function is itself.

               $$\int e^{u}\, du = e^{u}$$

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

         Now substitute $u$ back in:

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

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

2. Now simplify:

   $$\frac{e^{5 x - 7}}{5}$$

3. Add the constant of integration:

   $$\frac{e^{5 x - 7}}{5}+ \mathrm{constant}$$

The answer is:

$$\frac{e^{5 x - 7}}{5}+ \mathrm{constant}$$