Calculate the integral ∫ (sin(2x)) dx. What is the result?

Calculate the integral ∫ (sin(2x)) dx. What is the result?

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

Answer: (A)

To find the integral of sin(2x) with respect to x, we can apply a standard technique involving substitution or using the integration formula for sine functions. When integrating sin(kx), the general rule is: ∫ sin(kx) dx = - (1/k) cos(kx) + C In this context, k equals 2 for the function sin(2x). Therefore, we substitute k with 2: ∫ sin(2x) dx = - (1/2) cos(2x) + C This result indicates that for every increment of x, the area under the curve defined by sin(2x) is reflected as a negative cosine function scaled by 1/2, plus an arbitrary constant C. Thus, the correct result of the integral is indeed -(1/2) cos(2x) + C, which confirms the first choice provided in the question. Other options, which suggest different outcomes, do not adhere to the correct application of integration rules for sine functions.

To find the integral of sin(2x) with respect to x, we can apply a standard technique involving substitution or using the integration formula for sine functions.

When integrating sin(kx), the general rule is:

∫ sin(kx) dx = - (1/k) cos(kx) + C

In this context, k equals 2 for the function sin(2x). Therefore, we substitute k with 2:

∫ sin(2x) dx = - (1/2) cos(2x) + C

This result indicates that for every increment of x, the area under the curve defined by sin(2x) is reflected as a negative cosine function scaled by 1/2, plus an arbitrary constant C.

Thus, the correct result of the integral is indeed -(1/2) cos(2x) + C, which confirms the first choice provided in the question.

Other options, which suggest different outcomes, do not adhere to the correct application of integration rules for sine functions.