`int_(0)^(1)(1)/(sqrt(x+3))dx =`

Options

A. `4-2sqrt(3)`
B. `2sqrt(3)-4`
C. `4+2sqrt(3)`
D. `4sqrt(3)-2`

Answer

To solve the integral \( I = \int_{0}^{1} \frac{1}{\sqrt{x+3}} \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( z = x + 3 \). Then, differentiate both sides:
\[ dz = dx \]
Now, we need to change the limits of integration. When \( x = 0 \):
\[ z = 0 + 3 = 3 \]
When \( x = 1 \):
\[ z = 1 + 3 = 4 \]
Thus, the integral becomes:
\[ I = \int_{3}^{4} \frac{1}{\sqrt{z}} \, dz \]

### Step 2: Rewrite the Integral
The expression \( \frac{1}{\sqrt{z}} \) can be rewritten as \( z^{-1/2} \):
\[ I = \int_{3}^{4} z^{-1/2} \, dz \]

### Step 3: Integrate
Using the power rule for integration, where \( \int z^n \, dz = \frac{z^{n+1}}{n+1} + C \), we have:
\[ \int z^{-1/2} \, dz = \frac{z^{1/2}}{1/2} = 2z^{1/2} \]
Thus,
\[ I = \left[ 2\sqrt{z} \right]_{3}^{4} \]

### Step 4: Evaluate the Limits
Now, we substitute the limits into the integrated function:
\[ I = 2\sqrt{4} - 2\sqrt{3} \]
Calculating this gives:
\[ I = 2 \cdot 2 - 2\sqrt{3} = 4 - 2\sqrt{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{4 - 2\sqrt{3}} \]

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