If `I_(1)=int_(0)^(pi//4)sin^(2)xdx and I_(2)=int_(0)^(pi//4)cos^(2)xdx`, then

Options

A. `I_(1)=I_(2)`
B. `I_(1) lt I_(2)`
C. `I_(1) gt I_(2)`
D. `I_(2)=I_(1)+pi//4`

Answer

To solve the integrals \( I_1 = \int_0^{\frac{\pi}{4}} \sin^2 x \, dx \) and \( I_2 = \int_0^{\frac{\pi}{4}} \cos^2 x \, dx \), we can use the trigonometric identities and properties of definite integrals.

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

Using the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \), we can rewrite \( I_1 \):

\[ I_1 = \int_0^{\frac{\pi}{4}} \sin^2 x \, dx = \int_0^{\frac{\pi}{4}} \frac{1 - \cos(2x)}{2} \, dx \]

### Step 2: Split the Integral

Now, we can split the integral:

\[ I_1 = \frac{1}{2} \int_0^{\frac{\pi}{4}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{4}} \cos(2x) \, dx \]

### Step 3: Evaluate the First Integral

The first integral is straightforward:

\[ \int_0^{\frac{\pi}{4}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{4}} = \frac{\pi}{4} \]

### Step 4: Evaluate the Second Integral

For the second integral, we have:

\[ \int_0^{\frac{\pi}{4}} \cos(2x) \, dx = \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{4}} = \frac{\sin\left(\frac{\pi}{2}\right)}{2} - \frac{\sin(0)}{2} = \frac{1}{2} - 0 = \frac{1}{2} \]

### Step 5: Combine Results for \( I_1 \)

Now, substituting back into the expression for \( I_1 \):

\[ I_1 = \frac{1}{2} \left( \frac{\pi}{4} \right) - \frac{1}{2} \left( \frac{1}{2} \right) = \frac{\pi}{8} - \frac{1}{4} \]

### Step 6: Calculate \( I_2 \)

Using the identity \( \cos^2 x = \frac{1 + \cos(2x)}{2} \), we can rewrite \( I_2 \):

\[ I_2 = \int_0^{\frac{\pi}{4}} \cos^2 x \, dx = \int_0^{\frac{\pi}{4}} \frac{1 + \cos(2x)}{2} \, dx \]

### Step 7: Split the Integral for \( I_2 \)

Similar to \( I_1 \):

\[ I_2 = \frac{1}{2} \int_0^{\frac{\pi}{4}} 1 \, dx + \frac{1}{2} \int_0^{\frac{\pi}{4}} \cos(2x) \, dx \]

### Step 8: Evaluate the Integrals for \( I_2 \)

Using the results from \( I_1 \):

\[ I_2 = \frac{1}{2} \left( \frac{\pi}{4} \right) + \frac{1}{2} \left( \frac{1}{2} \right) = \frac{\pi}{8} + \frac{1}{4} \]

### Step 9: Compare \( I_1 \) and \( I_2 \)

Now we have:

\[ I_1 = \frac{\pi}{8} - \frac{1}{4}, \quad I_2 = \frac{\pi}{8} + \frac{1}{4} \]

### Step 10: Conclusion

Clearly, \( I_2 > I_1 \).

### Final Result

Thus, we can conclude that \( I_2 \) is greater than \( I_1 \).

---