Consider the function f(x)=9x3−4x5.
Let F(x) be the antiderivative of f(x) with F(1)=0.
Then F(4) =
f(x)=9x³-4x^5
F(x)=∫f(x)dx=(9/4)x^4-(4/6)x^6+C
Given F(1)=0, solve for C and hence
find F(4)
F(1)=(9/4)1^4-(4/6)1^6+C=0
=>
C=(4/6)-9/4=(8-27)/12=-19/12
=>
F(x)=(9/4)x^4-(4/6)x^6-19/12
Find
F(4)
To find the value of F(4), we need to find the antiderivative of f(x) and then evaluate it at x = 4.
The given function is f(x) = 9x^3 - 4x^5. To find the antiderivative of f(x), we need to find a function F(x) such that F'(x) = f(x).
To do that, we need to reverse the process of finding the derivative. We can use the power rule to find the antiderivative of each term of f(x). The power rule states that for any term x^n, the antiderivative is (1/(n+1)) * x^(n+1).
Using the power rule, we find:
∫ (9x^3 - 4x^5) dx = (9/4) * x^4 - (4/6) * x^6 + C
Now, to find the specific antiderivative F(x), we need to determine the constant of integration, denoted by C. We are given that F(1) = 0, so we can substitute x = 1 into the antiderivative expression and solve for C.
F(1) = (9/4) * 1^4 - (4/6) * 1^6 + C
0 = (9/4) - (4/6) + C
0 = 9/4 - 2/3 + C
0 = (27/12) - (8/12) + C
0 = 19/12 + C
To solve for C, we multiply both sides of the equation by 12:
0 = 19 + 12C
-19 = 12C
C = -19/12
Now that we have the value of C, we can substitute it into the antiderivative expression:
F(x) = (9/4) * x^4 - (4/6) * x^6 - 19/12
Finally, to find F(4), we substitute x = 4 into the expression for F(x):
F(4) = (9/4) * 4^4 - (4/6) * 4^6 - 19/12
Calculating this expression, we find the value of F(4).