For a certain drug, the rate of reaction in appropriate units is given by Upper R prime (t) equalsStartFraction 2 Over t plus 1 EndFraction plus StartFraction 1 Over Start Root t plus 1 End Root End Fraction where t is time (in hours) after the drug is administered. Find the total reaction to the drug over the following time periods.

a. From t=1 to t=12.

b. From t=12 to t=24

a) 8.13

b) 4.10

Step-by-step explanation:

Given the rate of reaction R' (t) = 2/t+1 + 1/√t+1

In order to get the total reaction R (t) to the drugs at this times, we need to first integrate the given function to get R (t)

On integrating R' (t)

∫ (2/t+1 + 1/√t+1) dt

In integration, k∫f' (x) / f (x) dx = 1/k ln (fx) + C where k is any constant.

∫ (2/t+1 + 1/√t+1) dt

= ∫ (2/t+1) dt + ∫ (1/√t+1) dt

= 2∫ 1/t+1 dt + ∫1 / + (t+1) ^1/2 dt

= 2ln (t+1) + 2 (t+1) ^1/2 + C

= 2ln (t+1) + 2√ (t+1) + C

a) For total reactions from t = 1 to t = 12

When t = 1

R (1) = 2ln2 + 2√2

≈ 4.21

When t = 12

R (12) = 2ln13 + 2√13

≈ 12.34

R (12) - R (1) ≈ 12.34-4.21

≈ 8.13

Total reactions to the drugs over the period from t = 1 to t = 12 is approx 8.13.

b) For total reactions from t = 12 to t = 24

When t = 12

R (12) = 2ln13 + 2√13

≈ 12.34

When t = 24

R (24) = 2ln25 + 2√25

≈ 16.44

R (12) - R (1) ≈ 16.44-12.34

≈ 4.10

Total reactions to the drugs over the period from t = 12 to t = 24 is approx 4.10