Consider the function f(x)=(7/x^2)-(6/x^6).
Let F(x) be the antiderivative of f(x) with F(1)=0.
Then F(2) equals _____.
Write it as
f(x) = 7x^-2 - 6x^-6
F(x)= -7x^-1 + (6/5)x^-5 + c
F(2) = -7/2 + (6/5)/2^5 + c
= ......
since F(1) = 0, we have
-7 + 6/5 + c = 0
which you can plug in to evaluate F(2)
To find F(x), the antiderivative of f(x), we will integrate the given function f(x):
∫[ (7/x^2) - (6/x^6) ] dx
To integrate the function, we treat each term separately.
∫(7/x^2) dx = 7 * ∫(1/x^2) dx = 7 * (-1/x) = -7/x
∫(6/x^6) dx = 6 * ∫(1/x^6) dx = 6 * (-1/5x^5) = -6/5x^5
So, the antiderivative becomes:
F(x) = -7/x - 6/5x^5 + C
Given that F(1) = 0, we can substitute the value of x = 1 into the equation:
0 = -7/1 - 6/5(1)^5 + C
Simplifying:
0 = -7 - 6/5 + C
Now, let's find the value of C:
7 - 6/5 + C = 0
Multiplying through by 5 to remove the fractional term:
35 - 6 + 5C = 0
29 + 5C = 0
5C = -29
C = -29/5
Now that we have the value of C, we can rewrite the antiderivative equation as:
F(x) = -7/x - 6/5x^5 - 29/5
To find F(2), we substitute x = 2 into the expression for F(x):
F(2) = -7/2 - 6/5(2)^5 - 29/5
Simplifying:
F(2) = -7/2 - 6/5(32) - 29/5
F(2) = -7/2 - 192/5 - 29/5
To add the fractions, we need a common denominator of 10:
F(2) = (-35/10) - (384/10) - (58/10)
F(2) = -477/10
Therefore, F(2) equals -477/10.
To find the value of F(2), we need to find the antiderivative of f(x) first.
The function f(x) = (7/x^2) - (6/x^6) can be rewritten as (7x^(-2)) - (6x^(-6)).
To find the antiderivative of f(x), we can use the power rule of integration, which states that the antiderivative of x^n with respect to x is (x^(n+1))/(n+1), except when n = -1 which gives ln|x|.
Applying the power rule to each term in f(x), the antiderivative F(x) becomes:
F(x) = (7x^(-2+1))/(1+1) - (6x^(-6+1))/(1+1)
= (7x^(-1))/(2) - (6x^(-5))/(2)
= (7/x) - (3/x^5)
Now, we know that F(1) = 0, so we can substitute x = 1 into the antiderivative F(x):
F(1) = (7/1) - (3/1)
= 7 - 3
= 4
Finally, to find F(2), we substitute x = 2 into the antiderivative F(x):
F(2) = (7/2) - (3/2)
= 7/2 - 3/2
= 4/2
= 2
Therefore, F(2) equals 2.