Q: Find the antiderivative F of f(x)=4−3(1+x^2)^−1 that satisfies F(1)=−9.
I got 4x-3arctan(x), but it still seems to be incorrect, and the hw doesn't care for constants (like +C at the end). Am I missing something?
you are correct as far as you went. Now just use the data point they gave you to find C:
4*1-3arctan(1) + C = -9
To find the antiderivative F of f(x) = 4 − 3(1 + x^2)^−1, we can start by trying to simplify the integral.
Given that F(1) = -9, we can use this information to determine the value of the constant of integration.
Let's begin by rewriting f(x) as:
f(x) = 4 - 3/(1 + x^2)
Now, to integrate f(x), we can split the integrand into two separate terms:
∫(4 - 3/(1 + x^2)) dx
The first term, 4, is a constant and therefore integrates to:
∫4 dx = 4x
For the second term, we can use the substitution method by letting u = 1 + x^2.
Differentiating u with respect to x, we get:
du/dx = 2x
Rearranging the equation, we can solve for dx:
dx = du / (2x)
Now, substituting the value of dx in terms of du into the integral:
∫3/(1 + x^2) dx = ∫3/u du
Since u = 1 + x^2:
∫3/(1 + x^2) dx = ∫3/u du = 3ln|u| + C
Substituting back for u, we get:
3ln|1 + x^2| + C
Now, adding the two integrated terms back together, we have:
F(x) = 4x + 3ln|1 + x^2| + C
Given that F(1) = -9, we can substitute x = 1 into the equation:
4(1) + 3ln|1 + (1)^2| + C = -9
Simplifying:
4 + 3ln(2) + C = -9
3ln(2) + C = -9 - 4
3ln(2) + C = -13
Now, we can solve for the value of the constant of integration C:
C = -13 - 3ln(2)
Therefore, the antiderivative F(x) of f(x) that satisfies F(1) = -9 is:
F(x) = 4x + 3ln|1 + x^2| - 13 - 3ln(2)
To find the antiderivative of f(x) = 4 - 3(1 + x^2)^(-1), we can use the power rule for integration, which states that ∫x^n dx = (1/(n+1))x^(n+1) + C, where C is the constant of integration.
Let's proceed step by step:
1. Start with the given function f(x) = 4 - 3(1 + x^2)^(-1).
2. Distribute the negative sign: f(x) = 4 - 3/(1 + x^2).
3. Separate the terms: f(x) = 4 - 3 + (-3/(1 + x^2)).
4. Combine like terms: f(x) = 1 - 3/(1 + x^2).
5. Now, we need to integrate this function. First, let's integrate the constant term: ∫1 dx = x + C_1, where C_1 is a constant.
6. For the second term, -3/(1 + x^2), we can rewrite it as -3(1 + x^2)^(-1) and apply the power rule.
Using the power rule, we have: ∫(1 + x^2)^(-1) dx = arctan(x) + C_2, where C_2 is another constant.
7. Therefore, the antiderivative of f(x) is F(x) = x - 3arctan(x) + C, where C = C_1 + C_2 is the constant of integration.
However, you mentioned that the homework doesn't care for constants at the end. In that case, you can simplify the answer as F(x) = x - 3arctan(x), since constant terms cancel out when calculating the difference between two antiderivatives.
Now, to find the value of the constant C, we can use the given condition F(1) = -9. Substituting x = 1 into F(x), we have:
F(1) = 1 - 3arctan(1) + C = 1 - 3(π/4) + C.
Since F(1) = -9, we can solve for C:
-9 = 1 - 3(π/4) + C.
C = -9 - 1 + 3(π/4) = (-10 + 3(π/4)).
Therefore, the antiderivative F(x) that satisfies F(1) = -9 is F(x) = x - 3arctan(x) + (-10 + 3(π/4)).