Integrate the following :
`int(7e^(5x)+7^(x)+3)dx.`

Options

A. `7(e^(5x))/(5)+(7^(x))/(log_(e)7)+3x.`
B. `7(e^(5x))+(7^(x))/(log_(e)7)+3x.`
C. `(e^(5x))/(5)+(7^(x))/(log_(e)7)+x.`
D. `7(e^(5x))/(5)+(7^(x))/(log_(e)7)+3.`

Explanation

To solve the integral \( \int (7e^{5x} + 7^x + 3) \, dx \), we can break it down into separate integrals. Let's go through the steps one by one.

### Step 1: Separate the Integral
We can express the integral as the sum of three separate integrals:
\[ \int (7e^{5x} + 7^x + 3) \, dx = \int 7e^{5x} \, dx + \int 7^x \, dx + \int 3 \, dx \]

### Step 2: Integrate Each Term
Now, we will integrate each term separately.

#### Integrate \( 7e^{5x} \)
Using the formula for integrating \( e^{ax} \):
\[ \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \]
Here, \( a = 5 \), so:
\[ \int 7e^{5x} \, dx = 7 \cdot \frac{1}{5} e^{5x} = \frac{7}{5} e^{5x} \]

#### Integrate \( 7^x \)
Using the formula for integrating \( a^x \):
\[ \int a^x \, dx = \frac{1}{\ln(a)} a^x + C \]
Here, \( a = 7 \), so:
\[ \int 7^x \, dx = \frac{1}{\ln(7)} 7^x \]

#### Integrate \( 3 \)
The integral of a constant is:
\[ \int c \, dx = cx + C \]
So:
\[ \int 3 \, dx = 3x \]

### Step 3: Combine the Results
Now, we can combine all the results from the three integrals:
\[ \int (7e^{5x} + 7^x + 3) \, dx = \frac{7}{5} e^{5x} + \frac{1}{\ln(7)} 7^x + 3x + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int (7e^{5x} + 7^x + 3) \, dx = \frac{7}{5} e^{5x} + \frac{1}{\ln(7)} 7^x + 3x + C \]