Choose the correct Answer of the Following Questions : `int_1^4 x^(-1/2) dx` is equal to

Options

A. -2
B. -7/16
C. 1/2
D. 2

Answer

To solve the integral \( \int_1^4 x^{-\frac{1}{2}} \, dx \), we will follow these steps:

### Step 1: Identify the integral form
The integral we need to solve is of the form \( \int x^n \, dx \), where \( n = -\frac{1}{2} \).

### Step 2: Apply the power rule for integration
According to the power rule for integration, we have:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For definite integrals, we do not include the constant \( C \).

### Step 3: Substitute \( n \) into the formula
Substituting \( n = -\frac{1}{2} \) into the formula gives:
\[ n + 1 = -\frac{1}{2} + 1 = \frac{1}{2} \]
Thus, we have:
\[ \int x^{-\frac{1}{2}} \, dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} \]

### Step 4: Evaluate the definite integral
Now we need to evaluate this from the limits 1 to 4:
\[ \int_1^4 x^{-\frac{1}{2}} \, dx = \left[ 2x^{\frac{1}{2}} \right]_1^4 \]
This means we will calculate:
\[ 2(4^{\frac{1}{2}}) - 2(1^{\frac{1}{2}}) \]

### Step 5: Calculate the values
Calculating the values:
- \( 4^{\frac{1}{2}} = \sqrt{4} = 2 \)
- \( 1^{\frac{1}{2}} = \sqrt{1} = 1 \)

So we have:
\[ 2(2) - 2(1) = 4 - 2 = 2 \]

### Conclusion
Thus, the value of the integral \( \int_1^4 x^{-\frac{1}{2}} \, dx \) is equal to \( 2 \).

### Final Answer
The correct answer is \( 2 \).

---