`int ( sin^(4) x - cos ^(4) x ) dx ` is equal to

Options

A. `(1)/(2) cos 2x +C`
B. `- ( 1)/(2) cos 2x +C`
C. ` (1)/(2) sin 2x + C`
D. ` - (1)/(2) sin 2x + C`

Answer

To solve the integral \( \int (\sin^4 x - \cos^4 x) \, dx \), we can follow these steps:

### Step 1: Use the difference of squares identity
We can rewrite \( \sin^4 x - \cos^4 x \) using the identity \( a^2 - b^2 = (a - b)(a + b) \):
\[ \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \]

### Step 2: Simplify using the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 x + \cos^2 x = 1 \]
Thus, we can simplify our expression:
\[ \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(1) = \sin^2 x - \cos^2 x \]

### Step 3: Substitute for \( \cos^2 x \)
Next, we can express \( \cos^2 x \) in terms of \( \sin^2 x \):
\[ \cos^2 x = 1 - \sin^2 x \]
Substituting this into our integral gives:
\[ \sin^2 x - \cos^2 x = \sin^2 x - (1 - \sin^2 x) = 2\sin^2 x - 1 \]

### Step 4: Rewrite the integral
Now, we can rewrite our integral as:
\[ \int (\sin^2 x - \cos^2 x) \, dx = \int (2\sin^2 x - 1) \, dx \]

### Step 5: Use the double angle identity
We can use the double angle identity for cosine, which states:
\[ \cos 2x = 1 - 2\sin^2 x \quad \text{or} \quad 2\sin^2 x - 1 = -\cos 2x \]
Thus, we can rewrite our integral as:
\[ \int (2\sin^2 x - 1) \, dx = \int -\cos 2x \, dx \]

### Step 6: Integrate
Now we can integrate:
\[ \int -\cos 2x \, dx = -\frac{1}{2} \sin 2x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ \int (\sin^4 x - \cos^4 x) \, dx = -\frac{1}{2} \sin 2x + C \]

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