Find the value of `int _( 0) ^(pi//4) (sin x) ^(4)` dx :

Options

A. `(3pi)/(16)`
B. `(3pi)/(32) -1/4`
C. `(3pi)/(32)-3/4`
D. `(3pi)/(16)-7/8`

Answer

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

### Step 1: Rewrite \( \sin^4 x \)
Using the identity for \( \sin^2 x \):
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
we can express \( \sin^4 x \) as:
\[ \sin^4 x = (\sin^2 x)^2 = \left(\frac{1 - \cos(2x)}{2}\right)^2 = \frac{(1 - \cos(2x))^2}{4} \]

### Step 2: Expand the expression
Now, we expand \( (1 - \cos(2x))^2 \):
\[ (1 - \cos(2x))^2 = 1 - 2\cos(2x) + \cos^2(2x) \]
Using the identity \( \cos^2(2x) = \frac{1 + \cos(4x)}{2} \), we can rewrite it as:
\[ 1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2} = 1 - 2\cos(2x) + \frac{1}{2} + \frac{\cos(4x)}{2} \]
Combining the terms gives:
\[ \frac{3}{2} - 2\cos(2x) + \frac{\cos(4x)}{2} \]

### Step 3: Substitute back into the integral
Now substituting back into the integral:
\[ \int_{0}^{\frac{\pi}{4}} \sin^4 x \, dx = \int_{0}^{\frac{\pi}{4}} \frac{1}{4} \left( \frac{3}{2} - 2\cos(2x) + \frac{\cos(4x)}{2} \right) \, dx \]
This simplifies to:
\[ \frac{1}{4} \int_{0}^{\frac{\pi}{4}} \left( \frac{3}{2} - 2\cos(2x) + \frac{\cos(4x)}{2} \right) \, dx \]

### Step 4: Integrate term by term
Now we can integrate each term separately:
1. \( \int_{0}^{\frac{\pi}{4}} \frac{3}{2} \, dx = \frac{3}{2} \cdot \frac{\pi}{4} = \frac{3\pi}{8} \)
2. \( \int_{0}^{\frac{\pi}{4}} -2\cos(2x) \, dx = -2 \cdot \frac{\sin(2x)}{2} \bigg|_{0}^{\frac{\pi}{4}} = -\sin\left(\frac{\pi}{2}\right) + \sin(0) = -1 + 0 = -1 \)
3. \( \int_{0}^{\frac{\pi}{4}} \frac{\cos(4x)}{2} \, dx = \frac{1}{2} \cdot \frac{\sin(4x)}{4} \bigg|_{0}^{\frac{\pi}{4}} = \frac{1}{8}(\sin(\pi) - \sin(0)) = 0 \)

### Step 5: Combine the results
Now, combining these results:
\[ \int_{0}^{\frac{\pi}{4}} \sin^4 x \, dx = \frac{1}{4} \left( \frac{3\pi}{8} - 1 + 0 \right) = \frac{1}{4} \left( \frac{3\pi}{8} - 1 \right) \]

### Final Result
Thus, the value of the integral is:
\[ \int_{0}^{\frac{\pi}{4}} \sin^4 x \, dx = \frac{3\pi}{32} - \frac{1}{4} \]