`int_(0)^(7)(1)/(sqrt(4+3x))dx=`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the definite integral \( \int_{0}^{7} \frac{1}{\sqrt{4 + 3x}} \, dx \), we will use a substitution method. Here are the steps:

### Step 1: Substitution
Let \( t = 4 + 3x \). Then, we differentiate both sides to find \( dx \):
\[ dt = 3 \, dx \quad \Rightarrow \quad dx = \frac{dt}{3} \]

### Step 2: Change the limits of integration
Now, we need to change the limits of integration according to our substitution:
- When \( x = 0 \):
\[ t = 4 + 3(0) = 4 \]
- When \( x = 7 \):
\[ t = 4 + 3(7) = 4 + 21 = 25 \]

So, the new limits of integration are from \( t = 4 \) to \( t = 25 \).

### Step 3: Rewrite the integral
Now we can rewrite the integral in terms of \( t \):
\[ \int_{0}^{7} \frac{1}{\sqrt{4 + 3x}} \, dx = \int_{4}^{25} \frac{1}{\sqrt{t}} \cdot \frac{dt}{3} = \frac{1}{3} \int_{4}^{25} t^{-1/2} \, dt \]

### Step 4: Integrate
Now we can integrate \( t^{-1/2} \):
\[ \int t^{-1/2} \, dt = 2t^{1/2} + C \]
Thus,
\[ \frac{1}{3} \int_{4}^{25} t^{-1/2} \, dt = \frac{1}{3} \left[ 2t^{1/2} \right]_{4}^{25} \]

### Step 5: Evaluate the definite integral
Now we evaluate the integral at the limits:
\[ = \frac{1}{3} \left[ 2(25^{1/2}) - 2(4^{1/2}) \right] = \frac{1}{3} \left[ 2(5) - 2(2) \right] = \frac{1}{3} \left[ 10 - 4 \right] = \frac{1}{3} \cdot 6 = 2 \]

### Final Answer
Thus, the value of the integral \( \int_{0}^{7} \frac{1}{\sqrt{4 + 3x}} \, dx \) is \( 2 \).

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