`int(sin^(4)x-cos^(4)x)dx=`

Options

A. `sinx-cosx`
B. `sinx.cosx+c`
C. `-sinx.cos x+c`
D. `cos 2x+c`

Answer

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

### Step 1: Rewrite the expression
We start by rewriting the expression \( \sin^4 x - \cos^4 x \) using the identity for the difference of squares:
\[ \sin^4 x - \cos^4 x = (\sin^2 x)^2 - (\cos^2 x)^2 = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) \]

### 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 the expression into the integral
Now we can substitute this back into the integral:
\[ \int (\sin^4 x - \cos^4 x) \, dx = \int (\sin^2 x - \cos^2 x) \, dx \]

### Step 4: Use the double angle identity
We can further simplify \( \sin^2 x - \cos^2 x \) using the double angle identity:
\[ \sin^2 x - \cos^2 x = -\cos(2x) \]
Thus, we have:
\[ \int (\sin^2 x - \cos^2 x) \, dx = \int -\cos(2x) \, dx \]

### Step 5: Integrate the expression
Now we can integrate:
\[ \int -\cos(2x) \, dx = -\frac{1}{2} \sin(2x) + C \]
where \( C \) is the constant of integration.

### Final Result
Therefore, 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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