`int_(0)^(pi^(2)//4) sin sqrt(x)dx` equals to

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi^2}{4}} \sin(\sqrt{x}) \, dx \), we can follow these steps:

### Step 1: Substitution
Let \( x = t^2 \). Then, differentiating both sides gives:
\[ dx = 2t \, dt \]

### Step 2: Change of Limits
Now we need to change the limits of integration. When \( x = 0 \):
\[ t = \sqrt{0} = 0 \]
When \( x = \frac{\pi^2}{4} \):
\[ t = \sqrt{\frac{\pi^2}{4}} = \frac{\pi}{2} \]

### Step 3: Substitute in the Integral
Substituting \( x \) and \( dx \) into the integral, we have:
\[ I = \int_{0}^{\frac{\pi}{2}} \sin(t) \cdot 2t \, dt = 2 \int_{0}^{\frac{\pi}{2}} t \sin(t) \, dt \]

### Step 4: Integration by Parts
Now we will use integration by parts. Let:
- \( u = t \) → \( du = dt \)
- \( dv = \sin(t) \, dt \) → \( v = -\cos(t) \)

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we get:
\[ \int t \sin(t) \, dt = -t \cos(t) \bigg|_{0}^{\frac{\pi}{2}} + \int \cos(t) \, dt \]

### Step 5: Evaluate the First Part
Evaluating the first part:
\[ -t \cos(t) \bigg|_{0}^{\frac{\pi}{2}} = -\left(\frac{\pi}{2} \cdot \cos\left(\frac{\pi}{2}\right) - 0 \cdot \cos(0)\right) = -\left(\frac{\pi}{2} \cdot 0 - 0\right) = 0 \]

### Step 6: Evaluate the Second Part
Now we evaluate the integral:
\[ \int \cos(t) \, dt = \sin(t) \bigg|_{0}^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1 \]

### Step 7: Combine Results
Putting it all together:
\[ \int t \sin(t) \, dt = 0 + 1 = 1 \]
Thus,
\[ I = 2 \cdot 1 = 2 \]

### Final Answer
The value of the integral is:
\[ \int_{0}^{\frac{\pi^2}{4}} \sin(\sqrt{x}) \, dx = 2 \]

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