Suppose `I_1=int_0^(pi/2)cos(pisin^2x)dx and I_2=int_0^(pi/2)cos(2pisin^2x)dx and I_3=int_0^(pi/2) cos(pi sinx)dx`, then

Options

A. (a) `I_(1)=0`
B. (b) `I_(2)+I_(3)=0`
C. (c) `I_(1)+I_(2)+I_(3)=0`
D. (d) `I_(2)=I_(3)`

Answer

To solve the problem, we need to evaluate the integrals \( I_1 \), \( I_2 \), and \( I_3 \) and analyze their relationships.

### Step 1: Evaluate \( I_1 \)

We start with:
\[ I_1 = \int_0^{\frac{\pi}{2}} \cos(\pi \sin^2 x) \, dx \]

Using King's rule, which states that:
\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx \]
we can rewrite \( I_1 \):
\[ I_1 = \int_0^{\frac{\pi}{2}} \cos(\pi \cos^2 x) \, dx \]

Now we add the two expressions for \( I_1 \):
\[ 2I_1 = \int_0^{\frac{\pi}{2}} \left( \cos(\pi \sin^2 x) + \cos(\pi \cos^2 x) \right) \, dx \]

### Step 2: Use the Cosine Addition Formula

Using the identity \( \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \):
\[ 2I_1 = \int_0^{\frac{\pi}{2}} 2 \cos\left(\frac{\pi}{2}\right) \cos(2x) \, dx \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), we have:
\[ 2I_1 = 0 \implies I_1 = 0 \]

### Step 3: Evaluate \( I_2 \)

Next, we evaluate:
\[ I_2 = \int_0^{\frac{\pi}{2}} \cos(2\pi \sin^2 x) \, dx \]
We can rewrite this using the identity \( \cos(2\pi \sin^2 x) = \cos(\pi(1 - \cos(2x))) \):
\[ I_2 = \int_0^{\frac{\pi}{2}} \cos(\pi(1 - \cos(2x))) \, dx \]

### Step 4: Change of Variables

Let \( t = 2x \), then \( dx = \frac{dt}{2} \) and the limits change from \( 0 \) to \( \pi \):
\[ I_2 = \frac{1}{2} \int_0^{\pi} \cos(\pi \cos t) \, dt \]

### Step 5: Apply Jack's Property

Using Jack's property:
\[ \int_0^{2A} f(x) \, dx = 2 \int_0^{A} f(x) \, dx \text{ if } f(2A - x) = f(x) \]
we can conclude:
\[ I_2 = -\frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(\pi \sin t) \, dt = -\frac{1}{2} I_3 \]

### Step 6: Relationship Between \( I_2 \) and \( I_3 \)

From the previous step, we have:
\[ I_2 + I_3 = 0 \implies I_2 = -I_3 \]

### Conclusion

We have found:
- \( I_1 = 0 \)
- \( I_2 + I_3 = 0 \)

Thus, the correct options are:
- \( I_1 = 0 \)
- \( I_2 + I_3 = 0 \)