Integrate the functions sin⁴x

To integrate the function \( \sin^4 x \), we can use a trigonometric identity and substitution. Here’s a step-by-step approach:

Step 1: Use the Power-Reducing Formula

First, we can apply the power-reducing formula for sine:

• \( \sin^2 x = \frac{1 - \cos(2x)}{2} \)

Thus, we can express \( \sin^4 x \) as:

• \( \sin^4 x = (\sin^2 x)^2 = \left(\frac{1 - \cos(2x)}{2}\right)^2 \)

Step 2: Expand the Expression

Now, expand the squared term:

• \( \sin^4 x = \frac{(1 - \cos(2x))^2}{4} = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4} \)

Step 3: Simplify Further

Next, we can use the power-reducing formula again for \( \cos^2(2x) \):

• \( \cos^2(2x) = \frac{1 + \cos(4x)}{2} \)

Substituting this back gives us:

• \( \sin^4 x = \frac{1}{4} \left(1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}\right) \)

Step 4: Combine and Prepare for Integration

Combining the terms results in:

• \( \sin^4 x = \frac{1}{4} \left(\frac{3}{2} - 2\cos(2x) + \frac{1}{2}\cos(4x)\right) \)

This simplifies to:

• \( \sin^4 x = \frac{3}{8} - \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x) \)

Step 5: Integrate Each Term

Now, we can integrate each term separately:

• \( \int \sin^4 x \, dx = \int \left(\frac{3}{8} - \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x)\right) dx \)

This results in:

• \( = \frac{3}{8}x - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C \)

Final Result

The integral of \( \sin^4 x \) is:

\( \int \sin^4 x \, dx = \frac{3}{8}x - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C \)