The integral form of exponential growth equation will be

Options

A. `N_(0)=N_(t)e^(rt)`
B. `N_(t)-N_(o)=e^(rt)`
C. `N_(t)=N_(o)e^(rt)`
D. `(N_(t))/(N_(o))=(1)/(e^(rt))`

Answer

To derive the integral form of the exponential growth equation, we follow these steps:

### Step 1: Understand the basic exponential growth equation
The basic differential equation for exponential growth is given by:
\[ \frac{dn}{dt} = r \cdot n \]
where:
- \( n \) is the population size,
- \( r \) is the intrinsic rate of natural growth,
- \( \frac{dn}{dt} \) is the rate of change of the population size with respect to time.

### Step 2: Separate the variables
To solve this differential equation, we can separate the variables \( n \) and \( t \):
\[ \frac{dn}{n} = r \, dt \]

### Step 3: Integrate both sides
Next, we integrate both sides:
\[ \int \frac{1}{n} \, dn = \int r \, dt \]
This gives us:
\[ \ln(n) = rt + C \]
where \( C \) is the constant of integration.

### Step 4: Solve for \( n \)
To solve for \( n \), we exponentiate both sides:
\[ n = e^{rt + C} \]
We can rewrite \( e^{C} \) as \( n_0 \) (the initial population size at \( t = 0 \)):
\[ n = n_0 e^{rt} \]

### Step 5: Final form of the equation
Thus, the integral form of the exponential growth equation is:
\[ n(t) = n_0 e^{rt} \]
where:
- \( n(t) \) is the population size at time \( t \),
- \( n_0 \) is the initial population size,
- \( r \) is the intrinsic growth rate,
- \( e \) is the base of the natural logarithm.