Consider the function f(x)=9/x^22−9/x^5.
Let F(x) be the antiderivative of f(x) with F(1)=0.
Then F(3) equals= ???
F(x) = ∫ f(x) dx = 9∫(1/x^22 - 1/x^5) dx = 9(-1/21 * 1/x^21 + 1/4 * 1/x^4) + C
since F(1) = 0, 9(-1/21 + 1/4)+C = 0, so C = -51/28
F(x) = 9(-1/21 * 1/x^21 + 1/4 * 1/x^4) - 51/28
Now find F(3)
To find the value of F(3), we need to integrate the function f(x) and evaluate the result at x = 3.
First, let's find the indefinite integral of f(x):
∫ [9/x^22 - 9/x^5] dx
We can split this into two separate integrals using the linearity property of integration:
∫ (9/x^22) dx - ∫ (9/x^5) dx
The integral of (9/x^22) can be found by adding 1 to the exponent and dividing by the new exponent:
∫ (9/x^22) dx = (9/((-21)+1)) * (1/x^((-21)+1))
= -9/21 * (1/x^21)
= (-3/7) * (1/x^21)
The integral of (9/x^5) can be found similarly:
∫ (9/x^5) dx = (9/(5+1)) * (1/x^(5+1))
= 9/6 * (1/x^6)
= (3/2) * (1/x^6)
Now, we can integrate each term separately:
∫ (9/x^22) dx - ∫ (9/x^5) dx
= (-3/7) * (1/x^21) - (3/2) * (1/x^6) + C
Here, C is the constant of integration.
Now, to find the specific value of F(x) at x = 3, we substitute the limits of integration:
F(3) = [(-3/7) * (1/3^21) - (3/2) * (1/3^6) + C] evaluated from x = 1 to x = 3
Since F(1) = 0, we can substitute this value:
0 = [(-3/7) * (1/1^21) - (3/2) * (1/1^6) + C]
Simplifying further gives:
0 = -3/7 - 3/2 + C
To solve for C, we can simplify the equation:
0 = -6/14 - 21/14 + C
0 = (-6 - 21)/14 + C
0 = -27/14 + C
C = 27/14
Now, we can substitute the value of C back into the expression for F(3):
F(3) = [(-3/7) * (1/3^21) - (3/2) * (1/3^6) + 27/14]
Finally, calculate this expression to find the value of F(3).
To find the value of F(3), we need to evaluate the antiderivative of f(x) with the given condition F(1) = 0.
The given function is f(x) = 9/x^22 - 9/x^5.
To find F(x), we integrate f(x) with respect to x.
∫f(x) dx = ∫(9/x^22 - 9/x^5) dx
To integrate, we split the integral into two parts:
∫(9/x^22) dx - ∫(9/x^5) dx
Now, let's integrate each part:
∫(9/x^22) dx = 9∫(x^(-22)) dx = 9 * (-1/21x^21) + C1 = -9/(21x^21) + C1
∫(9/x^5) dx = 9∫(x^(-5)) dx = 9 * (-1/4x^4) + C2 = -9/(4x^4) + C2
Putting it all together, we have:
F(x) = -9/(21x^21) + C1 - 9/(4x^4) + C2
Since F(1) = 0, we can substitute x = 1 into the equation and solve for C1:
0 = -9/(21(1)^21) + C1 - 9/(4(1)^4) + C2
0 = -9/21 + C1 - 9/4 + C2
0 = -3/7 + C1 - 9/4 + C2
To simplify, we combine the constants:
3/7 = C1 + 9/4 - C2
Now, plug in x = 3 into the equation for F(x):
F(3) = -9/(21(3)^21) + C1 - 9/(4(3)^4) + C2
F(3) = -9/(21(3)^21) + C1 - 9/(4(81)) + C2
Simplifying the exponents:
F(3) = -9/(21(3^21)) + C1 - 9/324 + C2
Since we don't have the specific values of C1 and C2, we cannot simplify further without additional information.