Final answer:
To find the value of the given integral, we can use a substitution. After performing the substitution and finding the new limits of integration, we can substitute the given information to get the final answer of -1/9.
Step-by-step explanation:
In order to find the value of the integral ∫₋₁⁰ x f(3 x²) d x, we can use a substitution u = 3x². The differential du is then du = 6x dx.
Now, we need to find the limits of integration for the new variable u. When x = -1, we have u = 3(-1)² = 3, and when x = 0, we have u = 3(0)² = 0. So, the new limits of integration are 0 to 3 for the variable u.
Substituting these values and the new differential, we have:
∫₋₁⁰ x f(3 x²) d x = ∫₀³ (1/6)(u/3) f(u) du = (1/18) ∫₀³ u f(u) du.
Given that ∫₀³ f(u) du = -2, we can substitute this value into the equation above:
∫₋₁⁰ x f(3 x²) d x = (1/18) ∫₀³ u f(u) du = (1/18)(-2) = -1/9.