Final answer:
To evaluate the integral ∫[7 to 9] x*f''(x) dx, we use integration by parts, with values provided for f'(x) and f(x) at x=7 and x=9. After calculation, the result of the integral is found to be 90.
Step-by-step explanation:
To find the value of the integral ∫[7 to 9] x · f''(x) dx, we can utilize the technique of integration by parts. Integration by parts is derived from the product rule of differentiation and can be represented as ∫ u dv = uv - ∫ v du, where u and v are functions of x.
In this case, we let u = x and dv = f''(x)dx. Then, we need to find du (which is dx) and v (which would be f'(x), since the integral of the second derivative f''(x) gives us the first derivative f'(x)).
Now applying the formula, we have:
- u = x
- du = dx
- dv = f''(x)dx
- v = f'(x)
Since we are given specific values for f'(x) at x = 7 and x = 9, we can calculate v using these values:
- v(7) = f'(7) = 17
- v(9) = f'(9) = 12
Therefore, applying integration by parts, we have:
∫[7 to 9] x · f''(x) dx = [x · f'(x)] |79 - ∫[7 to 9] f'(x)dx
Plugging in the values, we find:
[x · f'(x)] |79 = 9·17 - (7·12) = 153 - 84 = 69
We then need to find the value of ∫[7 to 9] f'(x)dx. This integral represents the change in f(x) from x=7 to x=9, which is f(9) - f(7) = (-14) - (7) = -21.
Finally, we combine the results:
69 - (-21) = 69 + 21 = 90
So, the answer to the integral is 90.