Consider the function f(x) sin x + cos x Select one: sin x cos x is an antiderivative of f(x)_ b. ~ sinx + cosx is an antiderivative of f(x) None of the other four answers is correct: sin X + cos X is an antiderivative of f(x) e. Sin x cos x is an antiderivative of f (x).
The Correct Answer and Explanation is:
The correct answer is: None of the other four answers is correct.
Explanation:
You are asked to determine which expression is an antiderivative of the function f(x)=sin(x)+cos(x)f(x) = \sin(x) + \cos(x)f(x)=sin(x)+cos(x).
Antiderivative of f(x)f(x)f(x):
To find the antiderivative of f(x)=sin(x)+cos(x)f(x) = \sin(x) + \cos(x)f(x)=sin(x)+cos(x), we need to integrate the function with respect to xxx.∫(sin(x)+cos(x)) dx\int (\sin(x) + \cos(x)) \, dx∫(sin(x)+cos(x))dx
We can apply the basic rules of integration:
- The antiderivative of sin(x)\sin(x)sin(x) is −cos(x)-\cos(x)−cos(x), because ddx(−cos(x))=sin(x)\frac{d}{dx} (-\cos(x)) = \sin(x)dxd(−cos(x))=sin(x).
- The antiderivative of cos(x)\cos(x)cos(x) is sin(x)\sin(x)sin(x), because ddx(sin(x))=cos(x)\frac{d}{dx} (\sin(x)) = \cos(x)dxd(sin(x))=cos(x).
Thus, the antiderivative of f(x)=sin(x)+cos(x)f(x) = \sin(x) + \cos(x)f(x)=sin(x)+cos(x) is:−cos(x)+sin(x)+C-\cos(x) + \sin(x) + C−cos(x)+sin(x)+C
Where CCC is the constant of integration.
Review of the Options:
- sin(x) cos(x) is an antiderivative of f(x)f(x)f(x): This is incorrect because the correct antiderivative does not involve a product of sine and cosine, but rather their individual integrals.
- sin(x) + cos(x) is an antiderivative of f(x)f(x)f(x): This is also incorrect because the derivative of sin(x)+cos(x)\sin(x) + \cos(x)sin(x)+cos(x) is cos(x)−sin(x)\cos(x) – \sin(x)cos(x)−sin(x), not sin(x)+cos(x)\sin(x) + \cos(x)sin(x)+cos(x).
- None of the other four answers is correct: This is the correct choice because none of the other given options match the actual antiderivative.
- sin(x) cos(x) is an antiderivative of f(x)f(x)f(x) (repeated): As explained above, this is incorrect.
Therefore, None of the other four answers is correct is the accurate answer. The correct antiderivative is −cos(x)+sin(x)+C-\cos(x) + \sin(x) + C−cos(x)+sin(x)+C.
