Give an example of a function F(x) that is an antiderivative of f(x) = 9 cos(9x)+ 3x^2

We are integrating f(x) = 9cos(9x) + 3x²:    

a) Apply the sum rule

b) Calculate each antiderivative

First integral

1. Take out the constant

2. Apply u-substitution, where u is 9x

3. Take out the constant (again)

4. Take the common integral of cos, which is sin

5. Substitute the original function back in for u and simplify

6. Always remember to add an arbitrary constant, C, at the end

Second integral

1. Take out the constant

2. Apply the power rule, , where a is your exponent

⇒

3. Add the arbitrary constant

c) Add the integrals

sin(9x) + C + x³ + C = sin(9x) + x³ + C

Notice the two arbitrary constants.  Since we do not know what either constant is, we can combine them into one arbitrary constant.

F(x) = sin(9x) + x³ + C

, choose a constant that is any number you want for C

Step-by-step explanation: