Question

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

Answer 1

`g(x)=sin(9x)+x^3+C`, choose a constant that is any number you want for C

Explanation:

1. So we have a function,
   `f(x)=9cos(9x)+3x^2`
2. An antiderivative is also called an integral, so to find the antiderivative of f(x) we find the integral
3. so we will say g(x) is equal to the integral (antiderivative) of f(x)
4. Then
   `g(x)=\int\ {(9cos(9x)+3x^(2)) } \, dx`
5. Which can be written as
   `g(x)=\int\ {9cos(9x)} \, dx +\int\ {3x^(2) } \, dx`
6. So let's do the first part integral
   `\int\ {9cos(9x)} \, dx` which can be written as
   `9*\int\ {cos(9x)} \, dx`
7. Then we use u substitution integration, and
   `u=9x,(du)/(dx) =9,dx=(1)/(9)*du`
8. so we replace dx and 9x so that
   `9*\int\ {cos(9x)} \, dx =9*\int\ {(1)/(9)cos(u) } \, du=9*(1)/(9)*\int\ {cos(u)} \, du`
9. Leaving
   `\int\ {cos(u)} \, du=sin(u)+C` then replace u with 9x, so
   `sin(9x)+C`
10. Now for the second part integration
    `\int\ {3x^2} \, dx =3 *\int\ {x^2} \, dx=3*(x^(2+1))/(2+1)+C=3*(x^(3))/(3)+C=x^3+C`
11. so
    `g(x)=sin(9x)+C+x^3+C=sin(9x)+x^3+C`
12. so
    `g(x)=sin(9x)+x^3+C` is a general solution antiderivative to f(x)

Answer 2

We are integrating f(x) = 9cos(9x) + 3x²:
`\int\ {9cos(9x)+3x^(2) } \, dx`

a) Apply the sum rule

`\int\9cos(9x)} \, dx +\int\ 3x^(2) } \, dx`

b) Calculate each antiderivative

First integral

`\int\ {9cos(9x)} \, dx`

1. Take out the constant

`9\int\ {cos(9x)} \, dx`

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

`9\int\ {cos(u)(1)/(9) } du`

3. Take out the constant (again)

`9*(1)/(9) \int{cos(u)} du`

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

`9*(1)/(9)sin(u)}`

5. Substitute the original function back in for u and simplify
`9*(1)/(9) sin(9x) = sin(9x)`

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

`sin(9x) + C`

Second integral

`\int3x^(2) } \, dx`

1. Take out the constant

`3\int{x^(2) } \, dx`

2. Apply the power rule,
`\int{x}^(a) \, dx =(x^(a+1) )/(a+1)`, where a is your exponent

⇒
`3*(x^(2+1) )/(2+1) = x^(3)`

3. Add the arbitrary constant

`x^(3) + C`

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.

Answer:

F(x) = sin(9x) + x³ + C