Compute ∫ sin(x) dx.

Compute ∫ sin(x) dx.

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

Answer: (A)

When you look for an antiderivative of sin x, you want a function whose derivative is sin x. The derivative of cos x is -sin x, so the function whose derivative is sin x is the negative of cos x. That gives ∫ sin x dx = -cos x + C, and you can check by differentiating: d/dx(-cos x) = sin x. The constant C represents any vertical shift of the antiderivative. The other options don’t work because their derivatives don’t produce sin x (for example, cos x differentiates to -sin x, sin x to cos x, and -sin x to -cos x).

When you look for an antiderivative of sin x, you want a function whose derivative is sin x. The derivative of cos x is -sin x, so the function whose derivative is sin x is the negative of cos x. That gives ∫ sin x dx = -cos x + C, and you can check by differentiating: d/dx(-cos x) = sin x. The constant C represents any vertical shift of the antiderivative. The other options don’t work because their derivatives don’t produce sin x (for example, cos x differentiates to -sin x, sin x to cos x, and -sin x to -cos x).