Find the antiderivative of the function f(x)=cos(2x)+5.
In order to find the antiderivative of the function we're given, we first have to study the general structure of the function. This function consists of a sum between cos(2x) and 5. Therefore, we have : F(x)=∫cos(2x)dx+∫5dx.
We will first focus on ∫cos(2x)dx. Let's solve this by substitution. Let g(x)=2x. We have : ∫cos(g(x))g'(x)dx=∫cos(u)du=sin(u)+C. Hence, ∫cos(2x)*2dx=sin(2x)+C. Thus, ∫cos(2x)=sin(2x)/2+C.
Let's now focus on ∫5dx. This one is fairly easy as we know how to integrate constants: We have ∫5dx = 5x+C.
Therefore, F(x)=(sin(2x)/2)+5x+c
TD
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