Answer :
To solve the integral \(\int 2 e^{-x / 3} \, dx\), we will follow these steps:
### Step 1: Identify the integrand
The integrand is \(2 e^{-x / 3}\).
### Step 2: Simplify the integral using a substitution if necessary
Let's make a substitution to simplify the integration. We'll set:
[tex]\[ u = -\frac{x}{3} \][/tex]
### Step 3: Compute the differential \(du\)
From the substitution, differentiate both sides with respect to \(x\):
[tex]\[ \frac{du}{dx} = -\frac{1}{3} \][/tex]
[tex]\[ du = -\frac{1}{3} dx \][/tex]
[tex]\[ dx = -3 \, du \][/tex]
### Step 4: Substitute into the integral
Substitute \(u = -\frac{x}{3}\) and \(dx = -3 \, du\) into the integral:
[tex]\[ \int 2 e^{-x / 3} \, dx = \int 2 e^u (-3 \, du) \][/tex]
This simplifies to:
[tex]\[ \int 2 e^u (-3 \, du) = -6 \int e^u \, du \][/tex]
### Step 5: Integrate \(e^u\)
The integral of \(e^u\) with respect to \(u\) is simply \(e^u\):
[tex]\[ -6 \int e^u \, du = -6 e^u \][/tex]
### Step 6: Substitute back in terms of \(x\)
Recall that \(u = -\frac{x}{3}\), so substitute back:
[tex]\[ -6 e^u = -6 e^{-\frac{x}{3}} \][/tex]
Thus, the integral of \(2 e^{-x / 3}\) with respect to \(x\) is:
[tex]\[ \int 2 e^{-x / 3} \, dx = -6 e^{-\frac{x}{3}} + C \][/tex]
where \(C\) is the constant of integration.
### Final Answer
[tex]\[ \boxed{-6 e^{-\frac{x}{3}} + C} \][/tex]
### Step 1: Identify the integrand
The integrand is \(2 e^{-x / 3}\).
### Step 2: Simplify the integral using a substitution if necessary
Let's make a substitution to simplify the integration. We'll set:
[tex]\[ u = -\frac{x}{3} \][/tex]
### Step 3: Compute the differential \(du\)
From the substitution, differentiate both sides with respect to \(x\):
[tex]\[ \frac{du}{dx} = -\frac{1}{3} \][/tex]
[tex]\[ du = -\frac{1}{3} dx \][/tex]
[tex]\[ dx = -3 \, du \][/tex]
### Step 4: Substitute into the integral
Substitute \(u = -\frac{x}{3}\) and \(dx = -3 \, du\) into the integral:
[tex]\[ \int 2 e^{-x / 3} \, dx = \int 2 e^u (-3 \, du) \][/tex]
This simplifies to:
[tex]\[ \int 2 e^u (-3 \, du) = -6 \int e^u \, du \][/tex]
### Step 5: Integrate \(e^u\)
The integral of \(e^u\) with respect to \(u\) is simply \(e^u\):
[tex]\[ -6 \int e^u \, du = -6 e^u \][/tex]
### Step 6: Substitute back in terms of \(x\)
Recall that \(u = -\frac{x}{3}\), so substitute back:
[tex]\[ -6 e^u = -6 e^{-\frac{x}{3}} \][/tex]
Thus, the integral of \(2 e^{-x / 3}\) with respect to \(x\) is:
[tex]\[ \int 2 e^{-x / 3} \, dx = -6 e^{-\frac{x}{3}} + C \][/tex]
where \(C\) is the constant of integration.
### Final Answer
[tex]\[ \boxed{-6 e^{-\frac{x}{3}} + C} \][/tex]
