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:

for ex: the reciprocal of 3/4 is 4/3. it is the opposite.

reciprocal definition:

a mathematical expression or function so related to another that their product is one; the quantity obtained by dividing the number one by a given quantity.

option 2 is correct.

step-by-step explanation:

we are given two inequality equation

x+y> 5 and -3x+2y< -2

first we draw the graph of both function using t-table method

equation: x+y> 5

for x=0, y=5

for x=-5, y=0

table:

x   :   0     5

y   :   5     0

test point (0,0) , 0> 5

false

equation: -3x+2y< -2

for x=0, y=-1

for x=2, y=4

table:

x   :   0     2

y   : -1       4

test point (0,0) , 0< -2

true

now we draw the graph. see the attachment for graph.

common region would be solution set.

option 2 is correct.