What is the integral of sin^2(x) dx?

What is the integral of sin^2(x) dx?

• A. (1/2)(x - sin(x)cos(x)) + C
• B. (1/4)(x - sin(2x)) + C
• C. (1/2)(x - cos^2(x)) + C
• D. sin(x)cos(x) + C

Answer: (A)

The integral of sin²(x) with respect to x can be effectively solved using a trigonometric identity. The identity states that sin²(x) can be expressed in terms of cosine: sin²(x) = (1 - cos(2x))/2. This transformation simplifies the integration process. Thus, when integrating sin²(x), we write: ∫sin²(x) dx = ∫(1 - cos(2x))/2 dx = (1/2) ∫(1 - cos(2x)) dx = (1/2) (∫1 dx - ∫cos(2x) dx) = (1/2)(x - (1/2)sin(2x)) + C = (1/2)(x - sin(2x)/2) + C = (1/4)(2x - sin(2x)) + C. This leads to the correct result: ∫sin²(x) dx = (1/4)(x - sin(2x)) + C. So, the correct answer to the integral of sin²(x) clearly aligns with option B. The other options do not offer the

The integral of sin²(x) with respect to x can be effectively solved using a trigonometric identity. The identity states that sin²(x) can be expressed in terms of cosine:

sin²(x) = (1 - cos(2x))/2.

This transformation simplifies the integration process.

Thus, when integrating sin²(x), we write:

∫sin²(x) dx = ∫(1 - cos(2x))/2 dx

= (1/2) ∫(1 - cos(2x)) dx

= (1/2) (∫1 dx - ∫cos(2x) dx)

= (1/2)(x - (1/2)sin(2x)) + C

= (1/2)(x - sin(2x)/2) + C

= (1/4)(2x - sin(2x)) + C.

This leads to the correct result:

∫sin²(x) dx = (1/4)(x - sin(2x)) + C.

So, the correct answer to the integral of sin²(x) clearly aligns with option B.

The other options do not offer the