The value of `int_(a)^(a+pi//2) (sin^(4) x + cos^(4) x)dx` is

Options

A. independent of a
B. `a(pi//2)^(2)`
C. `(3pi)/(8)`
D. `(3)/(8)pi a^(2)`

Answer

To solve the integral \( \int_{a}^{a+\frac{\pi}{2}} (\sin^4 x + \cos^4 x) \, dx \), we can follow these steps:

### Step 1: Simplify the integrand
We can use the identity \( \sin^4 x + \cos^4 x \) to simplify the expression. We know that:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x \]
Since \( \sin^2 x + \cos^2 x = 1 \), we have:
\[ \sin^4 x + \cos^4 x = 1 - 2\sin^2 x \cos^2 x \]
Using the double angle identity, \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2(2x) \), we can rewrite it as:
\[ \sin^4 x + \cos^4 x = 1 - \frac{1}{2} \sin^2(2x) \]

### Step 2: Set up the integral
Now we can rewrite the integral:
\[ \int_{a}^{a+\frac{\pi}{2}} (\sin^4 x + \cos^4 x) \, dx = \int_{a}^{a+\frac{\pi}{2}} \left(1 - \frac{1}{2} \sin^2(2x)\right) \, dx \]

### Step 3: Split the integral
We can split the integral into two parts:
\[ \int_{a}^{a+\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_{a}^{a+\frac{\pi}{2}} \sin^2(2x) \, dx \]

### Step 4: Calculate the first integral
The first integral is straightforward:
\[ \int_{a}^{a+\frac{\pi}{2}} 1 \, dx = \left[x\right]_{a}^{a+\frac{\pi}{2}} = \left(a + \frac{\pi}{2}\right) - a = \frac{\pi}{2} \]

### Step 5: Calculate the second integral
For the second integral, we can use the identity:
\[ \sin^2(2x) = \frac{1 - \cos(4x)}{2} \]
Thus,
\[ \int \sin^2(2x) \, dx = \int \frac{1 - \cos(4x)}{2} \, dx = \frac{1}{2} \left(x - \frac{1}{4} \sin(4x)\right) + C \]
Now, we evaluate:
\[ \int_{a}^{a+\frac{\pi}{2}} \sin^2(2x) \, dx = \frac{1}{2} \left[ x - \frac{1}{4} \sin(4x) \right]_{a}^{a+\frac{\pi}{2}} \]
Calculating this gives:
\[ = \frac{1}{2} \left[ \left(a + \frac{\pi}{2}\right) - \frac{1}{4} \sin(4(a + \frac{\pi}{2})) - a + \frac{1}{4} \sin(4a) \right] \]
Since \( \sin(4(a + \frac{\pi}{2})) = -\cos(4a) \), we have:
\[ = \frac{1}{2} \left[ \frac{\pi}{2} + \frac{1}{4} \cos(4a) - \frac{1}{4} \sin(4a) \right] \]

### Step 6: Combine results
Now we combine our results:
\[ \int_{a}^{a+\frac{\pi}{2}} (\sin^4 x + \cos^4 x) \, dx = \frac{\pi}{2} - \frac{1}{2} \cdot \frac{1}{2} \left[ \frac{\pi}{2} + \frac{1}{4} \cos(4a) - \frac{1}{4} \sin(4a) \right] \]
This simplifies to:
\[ = \frac{\pi}{2} - \frac{1}{4} \left[ \frac{\pi}{2} + \frac{1}{4} \cos(4a) - \frac{1}{4} \sin(4a) \right] \]

### Final Result
After simplifying, we arrive at the final value of the integral.