f(x) = [3/x^2] - [5/x^7], F(x) is the antiderivative of f(x) with F(1)=0.
F(x) = ?
I have no idea how to approach this problem. The fractions are scaring me.
This is really pretty simple, I can't imagine how you are stuck.
INTf(x)= INT 3x^-2 dx -INT (5x^-7)dx
= -3x^-1 +5/6 x^-6 +C
F(1)=0 = -3+5/6+C=1
solve for C.
F(x) = -3x^-1 +5/6 x^-6 check my work, it is easy to make a typo
No worries! I'll walk you through how to approach this problem step by step.
To find the antiderivative, or integral, of f(x), we will need to evaluate the integral of each term separately. Let's break it down:
f(x) = [3/x^2] - [5/x^7]
The first term, [3/x^2], can be written as 3x^(-2).
The second term, [5/x^7], can be written as 5x^(-7).
Now, we'll integrate each term separately:
∫ (3x^(-2)) dx - ∫ (5x^(-7)) dx
To integrate the first term, we add 1 to the power and divide by the new power:
∫ (3x^(-2)) dx = 3 * (x^(-1)) / (-1) + C1
Simplifying, we get:
∫ (3x^(-2)) dx = -3/x + C1
For the second term, we apply the same integration rule:
∫ (5x^(-7)) dx = 5 * (x^(-6)) / (-6) + C2
Simplifying, we get:
∫ (5x^(-7)) dx = -5/(6x^6) + C2
Now, we combine the results from the two integrals:
F(x) = -3/x + C1 - 5/(6x^6) + C2
Since we know that F(1) = 0, we can substitute x = 1 into the equation and solve for C1 and C2:
0 = -3/1 + C1 - 5/(6 * 1^6) + C2
0 = -3 + C1 - 5/6 + C2
To simplify further, we can combine like terms:
0 = -3 - 5/6 + C1 + C2
To satisfy the equation, the constant terms need to cancel out. Therefore, we have:
C1 + C2 = 3 + 5/6
Since C1 and C2 are arbitrary constants, we can combine them into a single constant:
C = C1 + C2
Therefore:
C = 3 + 5/6
Now, we can rewrite the equation for F(x) using the combined constant C:
F(x) = -3/x - 5/(6x^6) + C
And that's the antiderivative of f(x), F(x), with F(1) = 0.