Evaluate `inte^(3x+4)dx`.

Options

A. `(e^(3x+4))/(4)`
B. `(e^(3x+4))/(3)`
C. `(e^(3x+4))`
D. `(e^(x))/(3)`

Explanation

To evaluate the integral \( \int e^{3x + 4} \, dx \), we will use the substitution method. Here are the steps:

### Step 1: Choose a substitution
Let \( t = 3x + 4 \). This substitution simplifies the exponent in the integrand.

### Step 2: Differentiate the substitution
Now, we differentiate \( t \) with respect to \( x \):
\[ dt = 3 \, dx \]
From this, we can express \( dx \) in terms of \( dt \):
\[ dx = \frac{dt}{3} \]

### Step 3: Substitute in the integral
Now, we substitute \( t \) and \( dx \) into the integral:
\[ \int e^{3x + 4} \, dx = \int e^t \cdot \frac{dt}{3} \]
This simplifies to:
\[ \frac{1}{3} \int e^t \, dt \]

### Step 4: Integrate
The integral of \( e^t \) is simply \( e^t \):
\[ \frac{1}{3} \int e^t \, dt = \frac{1}{3} e^t + C \]
where \( C \) is the constant of integration.

### Step 5: Substitute back to original variable
Now we substitute back \( t = 3x + 4 \):
\[ \frac{1}{3} e^{3x + 4} + C \]

### Final Answer
Thus, the evaluated integral is:
\[ \int e^{3x + 4} \, dx = \frac{1}{3} e^{3x + 4} + C \]

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