Problem 24
In Exercises \(21-24,\) use tabular integration to find the antiderivative. $$\int x^{3} \cos 2 x d x$$
Problem 25
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{(1-x)^{2}}$$
Problem 25
Radon-222 The decay equation for radon- 222 gas is known to be \(y=y_{0} e^{-0.18 t},\) with \(t\) in days. About how long will it take the amount of radon in a sealed sample of air to decay to 90\(\%\) of its original value?
Problem 25
In Exercises \(25-28\) , evaluate the integral analytically. Support your answer using NINT. $$\int_{0}^{\pi / 2} x^{2} \sin 2 x d x$$
Problem 25
In Exercises \(23-26,\) the logistic equation describes the growth of a population \(P,\) where \(t\) is measured in years. In each case, find (a) the carrying capacity of the population, (b) the size of the population when it is growing the fastest, and (c) the rate at which the population is growing when it is growing the fastest. $$\frac{d P}{d t}=0.0002 P(1200-P)$$
Problem 26
Polonium-21O The number of radioactive atoms remaining after \(t\) days in a sample of polonium- 210 that starts with \(y_{0}\) radioactive atoms is \(y=y_{0} e^{-0.005 t}\) (a) Find the element's half-life. (b) Your sample will not be useful to you after 95\(\%\) of the radioactive nuclei present on the day the sample arrives have disintegrated. For about how many days after the sample arrives will you be able to use the polonium?
Problem 26
In Exercises \(23-26,\) the logistic equation describes the growth of a population \(P,\) where \(t\) is measured in years. In each case, find (a) the carrying capacity of the population, (b) the size of the population when it is growing the fastest, and (c) the rate at which the population is growing when it is growing the fastest. $$\frac{d P}{d t}=10^{-5} P(5000-P)$$
Problem 26
In Exercises \(25-28\) , evaluate the integral analytically. Support your answer using NINT. $$\int_{0}^{\pi / 2} x^{3} \cos 2 x d x$$
Problem 26
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \sec ^{2}(x+2) d x$$
Problem 27
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \sqrt{\tan x} \sec ^{2} x d x$$
