Which of the following are antiderivatives of f(x)=sinxcosx?
I f(x)=(sin^2* x)/2
II f(x)=(cos^2* x)/2
III f(x)=-cos(2x)/4
To determine which of the given functions are antiderivatives of f(x) = sin(x)cos(x), we can differentiate each function and see if it matches f(x).
Let's start with function I: f(x) = (sin^2(x))/2.
Taking the derivative of f(x):
f'(x) = (1/2)(2sin(x)cos(x))
= sin(x)cos(x)
The derivative of f(x) matches the given function f(x). Therefore, I is an antiderivative of f(x).
Now let's move on to function II: f(x) = (cos^2(x))/2.
Taking the derivative of f(x):
f'(x) = (1/2)(-2sin(x)cos(x))
= -sin(x)cos(x)
The derivative of f(x) does not match the given function f(x). Thus, II is not an antiderivative of f(x).
Finally, let's consider function III: f(x) = -cos(2x)/4.
Taking the derivative of f(x):
f'(x) = -1/4 * d(cos(2x))/dx
= -1/4 * (-2sin(2x))
= 1/2 * sin(2x)
The derivative of f(x) does not match the given function f(x). Therefore, III is not an antiderivative of f(x).
In conclusion, the only antiderivative of f(x) = sin(x)cos(x) among the given functions is I: f(x) = (sin^2(x))/2.
To determine which of the given options are antiderivatives of f(x) = sin(x)cos(x), we need to take the derivative of each option and check if it matches f(x).
Let's start with option I: f(x) = (sin^2 x)/2.
To take the derivative, we can use the chain rule:
f'(x) = (2sinx)(cosx)/2 = sin(x)cos(x).
Since the derivative of option I matches f(x), option I is an antiderivative of f(x).
Now let's consider option II: f(x) = (cos^2 x)/2.
Taking the derivative, we once again use the chain rule:
f'(x) = (2cosx)(-sinx)/2 = -sin(x)cos(x).
Since the derivative of option II does not match f(x), option II is not an antiderivative of f(x).
Lastly, let's examine option III: f(x) = -cos(2x)/4.
Taking the derivative:
f'(x) = (2sin(2x))/4 = (1/2)sin(2x).
Since the derivative of option III does not match f(x), option III is also not an antiderivative of f(x).
Therefore, the only antiderivative of f(x) = sin(x)cos(x) among the given options is option I: f(x) = (sin^2 x)/2.
1 and 3
Why don't you in your head take the derivate of the answers?
d sin^2 x/2= 1/2 sinx cosx * 2