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

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$e^{5 x + 7} \cdot 1 = e^{7} e^{5 x}$$

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

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

      1. There are multiple ways to do this integral.

         Method #1

         1. Let $u = 5 x$.

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

            $$\int \frac{e^{u}}{25}\, du$$

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

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

               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:

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

         Method #2

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

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

            $$\int \frac{1}{25}\, du$$

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

               $$\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}$$

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

                  $$\int 1\, du = u$$

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

            Now substitute $u$ back in:

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

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

   Method #2

   1. Rewrite the integrand:

      $$e^{5 x + 7} \cdot 1 = e^{7} e^{5 x}$$

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

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

      1. Let $u = 5 x$.

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

         $$\int \frac{e^{u}}{25}\, du$$

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

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

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

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:

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