The value of `int_(1)^(4) e^(sqrt(x))dx`, is

Options

A. `e^(2)`
B. `2e^(2)`
C. `4e^(2)`
D. `3e^(2)`

Answer

To solve the integral \( I = \int_{1}^{4} e^{\sqrt{x}} \, dx \), we will use a substitution method.

### Step-by-step Solution:

1. **Substitution**: Let \( t = \sqrt{x} \). Then, \( x = t^2 \). We differentiate to find \( dx \):
\[ dx = 2t \, dt \]

2. **Change of Limits**: When \( x = 1 \), \( t = \sqrt{1} = 1 \). When \( x = 4 \), \( t = \sqrt{4} = 2 \). Thus, the limits change from \( x = 1 \) to \( x = 4 \) to \( t = 1 \) to \( t = 2 \).

3. **Rewrite the Integral**: Substitute \( x \) and \( dx \) in the integral:
\[ I = \int_{1}^{2} e^{t} \cdot 2t \, dt = 2 \int_{1}^{2} t e^{t} \, dt \]

4. **Integration by Parts**: To evaluate \( \int t e^{t} \, dt \), we use integration by parts. Let:
- \( u = t \) (thus \( du = dt \))
- \( dv = e^{t} dt \) (thus \( v = e^{t} \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):
\[ \int t e^{t} \, dt = t e^{t} - \int e^{t} \, dt \]
\[ = t e^{t} - e^{t} + C \]

5. **Evaluate the Integral**: Now we can substitute back into our integral:
\[ I = 2 \left[ t e^{t} - e^{t} \right]_{1}^{2} \]

6. **Substituting the Limits**: Calculate the definite integral:
\[ I = 2 \left[ (2 e^{2} - e^{2}) - (1 e^{1} - e^{1}) \right] \]
\[ = 2 \left[ (2 e^{2} - e^{2}) - (e - e) \right] \]
\[ = 2 \left[ e^{2} - 0 \right] = 2 e^{2} \]

### Final Result:
Thus, the value of the integral \( \int_{1}^{4} e^{\sqrt{x}} \, dx \) is:
\[ \boxed{2e^{2}} \]