Answer:
z=7
Step-by-step explanation:
If z represents 1/2 of 14, then z=14*1/2=14/2=7
Substitute
and
. Then the integral transforms to
![\displaystyle \int \frac{x\,dx}{(x^2+4)^3} = \frac12 \int \frac{du}{u^3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bx%5C%2Cdx%7D%7B%28x%5E2%2B4%29%5E3%7D%20%3D%20%5Cfrac12%20%5Cint%20%5Cfrac%7Bdu%7D%7Bu%5E3%7D)
Apply the power rule.
![\displaystyle \int \frac{du}{u^3} = -\dfrac1{2u^2} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bdu%7D%7Bu%5E3%7D%20%3D%20-%5Cdfrac1%7B2u%5E2%7D%20%2B%20C)
Now put the result back in terms of
.
![\displaystyle \int \frac{x\,dx}{(x^2+4)^3} = \frac12 \left(-\dfrac1{2u^2} + C\right) = -\dfrac1{4u^2} + C = \boxed{-\dfrac1{4(x^2+4)^2} + C}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%5Cfrac%7Bx%5C%2Cdx%7D%7B%28x%5E2%2B4%29%5E3%7D%20%3D%20%5Cfrac12%20%5Cleft%28-%5Cdfrac1%7B2u%5E2%7D%20%2B%20C%5Cright%29%20%3D%20-%5Cdfrac1%7B4u%5E2%7D%20%2B%20C%20%3D%20%5Cboxed%7B-%5Cdfrac1%7B4%28x%5E2%2B4%29%5E2%7D%20%2B%20C%7D)
The farther to the left the number is the greater it's value becomes
Present value of the amount she will earn while on sabbatical if the interest rate is 6% is $126,964.34
The value of money reduces as time passes, this is due to the interest factor. $1 is worth more today than tomorrow this is because of the time value of money. The present value of cash flows is discounted value of future cash flows. Discount factors at a given rate of interest are used to find out the present value of cash flows.
Here, the professor will receive $70,000 every 7 years for 42 years.
Hence he will receive $70,000 sabbatical in the 7th, 14th,21st,28th,35th, and 42nd years.
We will find the present value of such receipts and add them.
R=0.06
F=$70,000
![Present~Value = \displaystyle\frac{FV}{(1+r)^7}+\frac{FV}{(1+r)^{14}}+\frac{FV}{(1+r)^{21}}+\frac{FV}{(1+r)^{28}}+\frac{FV}{(1+r)^{35}}+\frac{FV}{(1+r)^{42}}](https://tex.z-dn.net/?f=Present~Value%20%3D%20%5Cdisplaystyle%5Cfrac%7BFV%7D%7B%281%2Br%29%5E7%7D%2B%5Cfrac%7BFV%7D%7B%281%2Br%29%5E%7B14%7D%7D%2B%5Cfrac%7BFV%7D%7B%281%2Br%29%5E%7B21%7D%7D%2B%5Cfrac%7BFV%7D%7B%281%2Br%29%5E%7B28%7D%7D%2B%5Cfrac%7BFV%7D%7B%281%2Br%29%5E%7B35%7D%7D%2B%5Cfrac%7BFV%7D%7B%281%2Br%29%5E%7B42%7D%7D)
![Present~Value = \displaystyle\frac{70,000}{(1.06)^7}+\frac{70,000}{(1.06)^{14}}+\frac{70,000}{(1.06)^{21}}+\frac{70,000}{(1.06)^{28}}+\frac{70,000}{(1.06)^{35}}+\frac{70,000}{(1.06)^{42}}](https://tex.z-dn.net/?f=Present~Value%20%3D%20%5Cdisplaystyle%5Cfrac%7B70%2C000%7D%7B%281.06%29%5E7%7D%2B%5Cfrac%7B70%2C000%7D%7B%281.06%29%5E%7B14%7D%7D%2B%5Cfrac%7B70%2C000%7D%7B%281.06%29%5E%7B21%7D%7D%2B%5Cfrac%7B70%2C000%7D%7B%281.06%29%5E%7B28%7D%7D%2B%5Cfrac%7B70%2C000%7D%7B%281.06%29%5E%7B35%7D%7D%2B%5Cfrac%7B70%2C000%7D%7B%281.06%29%5E%7B42%7D%7D)
=$126,964.34
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