`u=int_0^(pi/2)cos((2pi)/3sin^2x)dx` and `v=int_0^(pi/2) cos(pi/3 sinx) dx`

Options

A. 2u = v
B. 2u = 3v
C. u = v
D. u = 2v

Explanation

To solve the integrals \( u \) and \( v \) defined as:

\[ u = \int_0^{\frac{\pi}{2}} \cos\left(\frac{2\pi}{3} \sin^2 x\right) dx \]
\[ v = \int_0^{\frac{\pi}{2}} \cos\left(\frac{\pi}{3} \sin x\right) dx \]

we will follow these steps:

### Step 1: Use the property of definite integrals
We can use the property of definite integrals, which states that:

\[ \int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx \]

For \( u \), we apply this property:

\[ u = \int_0^{\frac{\pi}{2}} \cos\left(\frac{2\pi}{3} \sin^2 x\right) dx = \int_0^{\frac{\pi}{2}} \cos\left(\frac{2\pi}{3} \cos^2 x\right) dx \]

### Step 2: Add the two expressions for \( u \)
Now we can add the two expressions for \( u \):

\[ 2u = \int_0^{\frac{\pi}{2}} \cos\left(\frac{2\pi}{3} \sin^2 x\right) dx + \int_0^{\frac{\pi}{2}} \cos\left(\frac{2\pi}{3} \cos^2 x\right) dx \]

### Step 3: Combine the integrals
Since both integrals have the same limits, we can combine them:

\[ 2u = \int_0^{\frac{\pi}{2}} \left( \cos\left(\frac{2\pi}{3} \sin^2 x\right) + \cos\left(\frac{2\pi}{3} \cos^2 x\right) \right) dx \]

### Step 4: Use the cosine addition formula
Using the cosine addition formula, we know that:

\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

Let \( A = \frac{2\pi}{3} \sin^2 x \) and \( B = \frac{2\pi}{3} \cos^2 x \):

\[ 2u = \int_0^{\frac{\pi}{2}} 2 \cos\left(\frac{2\pi}{3} \cdot \frac{1}{2}\right) \cos\left(\frac{2\pi}{3} \cdot \left(\sin^2 x - \cos^2 x\right)\right) dx \]

### Step 5: Simplify the integral
Since \( \sin^2 x + \cos^2 x = 1 \), we can simplify:

\[ 2u = 2 \cos\left(\frac{\pi}{3}\right) \int_0^{\frac{\pi}{2}} \cos\left(\frac{2\pi}{3} \left(\cos 2x\right)\right) dx \]

### Step 6: Change of variable
Let \( t = 2x \), then \( dx = \frac{dt}{2} \):

\[ 2u = 2 \cos\left(\frac{\pi}{3}\right) \cdot \frac{1}{2} \int_0^{\pi} \cos\left(\frac{2\pi}{3} \cos t\right) dt \]

### Step 7: Relate \( u \) and \( v \)
Notice that \( v \) is defined as:

\[ v = \int_0^{\frac{\pi}{2}} \cos\left(\frac{\pi}{3} \sin x\right) dx \]

By the earlier steps, we find that:

\[ 2u = v \]

Thus, we conclude:

\[ u = \frac{1}{2} v \]

### Final Result
Therefore, the relationship between \( u \) and \( v \) is:

\[ u = \frac{1}{2} v \]