what is the antiderivative of 1/4sinx+3cosx
(1/4) sin x + 3 cos x
-(1/4)cos x + 3 sin x + constant
To find the antiderivative of the function (1/4)sin(x) + 3cos(x), we can use the basic rules of integration. The antiderivative, or the integral, represents the reverse process of differentiation, so we need to look for a function whose derivative matches the given function.
1. Recall that the antiderivative of sin(x) is -cos(x). Therefore, the antiderivative of (1/4)sin(x) would be (1/4)(-cos(x)) = -(1/4)cos(x).
2. The antiderivative of cos(x) is sin(x). Hence, the antiderivative of 3cos(x) is 3sin(x).
3. Finally, combining the two antiderivatives, we have -(1/4)cos(x) + 3sin(x) + C as the antiderivative of the given function, where C is the constant of integration.
In summary, the antiderivative of (1/4)sin(x) + 3cos(x) is -(1/4)cos(x) + 3sin(x) + C.