`int7e^(7x+5)dx`=

Options

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

Answer

To solve the integral \( \int 7 e^{7x + 5} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the expression inside the integral:
\[ \int 7 e^{7x + 5} \, dx = \int 7 e^{7x} e^{5} \, dx \]
Since \( e^5 \) is a constant, we can factor it out of the integral.

### Step 2: Factor Out the Constant
Now, we can factor out the constant \( 7 e^5 \):
\[ = 7 e^5 \int e^{7x} \, dx \]

### Step 3: Integrate the Exponential Function
Next, we need to integrate \( e^{7x} \). The integral of \( e^{nx} \) is given by:
\[ \int e^{nx} \, dx = \frac{e^{nx}}{n} + C \]
In our case, \( n = 7 \), so we have:
\[ \int e^{7x} \, dx = \frac{e^{7x}}{7} + C \]

### Step 4: Substitute Back into the Integral
Now we substitute this result back into our expression:
\[ = 7 e^5 \left( \frac{e^{7x}}{7} + C \right) \]

### Step 5: Simplify the Expression
The \( 7 \) in the numerator and denominator cancels out:
\[ = e^5 e^{7x} + 7 e^5 C \]
We can simplify this to:
\[ = e^{7x + 5} + C' \]
where \( C' = 7 e^5 C \) is just another constant.

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

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