Use integration by parts to find the given integral:
∫_9^1 (7t - 42)e^(7 - t)
= [-28(1 + e^8)] / e^2
Is my answer correct?
or... is it..
= -70e^(-2) + 14e^6
Is your 9^1 a misprint? What is the purpose of the 1? Why is there no "dt" in your integral?
Both of your answers are incorrect, since they are constants. The indefinite integral would be a function of t, and the definite integral would have to have specified limits of integration. You did not mention any.
it's 1^9 and I forgot to include dt... and I need to evaluate it when f(9) - f(1) and that's what my answer represents..
Let's work through the problem to find the correct answer.
To evaluate the integral using integration by parts, we need to apply the formula:
∫ u * dv = uv - ∫ v * du
In this case, we can assign:
u = 7t - 42 (function to differentiate)
dv = e^(7 - t) dt (function to integrate)
Taking the derivatives and integrals:
du = 7 dt
v = ∫ e^(7 - t) dt = -e^(7 - t)
Now, we can substitute these values into the integration by parts formula:
∫ (7t - 42)e^(7 - t) dt = (7t - 42) * (-e^(7 - t)) - ∫ (-e^(7 - t)) * 7 dt
Simplifying the first term:
= -7t * e^(7 - t) + 42 * e^(7 - t) + ∫ 7e^(7 - t) dt
Now we have a simpler integral to compute: ∫ 7e^(7 - t) dt
We can rewrite this as -7∫ e^(7 - t) dt, and integrate directly:
= -7 * (-e^(7 - t)) + C
= 7e^(7 - t) + C
Now, let's substitute the limits of integration, which are from 9 to 1:
= 7e^(7 - 1) - 7e^(7 - 9)
= 7e^6 - 7e^(-2)
So the correct answer is:
∫_9^1 (7t - 42)e^(7 - t) dt = 7e^6 - 7e^(-2)
Therefore, your second answer is correct:
= -70e^(-2) + 14e^6