Problem 48
In Exercises \(45-48\) , use Euler's Method with increments of \(\Delta x=-0.1\) to approximate the value of \(y\) when \(x=1.7\) \(\frac{d y}{d x}=x-2 y\) and \(y=1\) when \(x=2\)
Problem 48
In Exercises \(47-50,\) use integration by parts to establish the reduction formula. $$\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x$$
Problem 48
More on Repeated Linear Factors The Heaviside Method is not very effective at finding the unknown numerators for par- tial fraction decompositions with repeated linear factors, but here is another way to find them. (a) If \(\frac{x^{2}+3 x+5}{(x-1)^{3}}=\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{(x-1)^{3}},\) show that \(A(x-1)^{2}+B(x-1)+C=x^{2}+3 x+5\) (b) Expand and equate coefficients of like terms to show that \(A=1,-2 A+B=3,\) and \(A-B+C=5 .\) Then find \(A, B\) , (c) Use partial fractions to evaluate \(\int \frac{x^{2}+3 x+5}{(x-1)^{3}} d x\)
Problem 49
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int 2 \sin ^{2} x d x, \quad \cos 2 x=2 \sin ^{2} x-1$$
Problem 49
In Exercises \(47-50,\) use integration by parts to establish the reduction formula. $$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x, \quad a \neq 0$$
Problem 49
Multiple Choice A bank account earning continuously compounded interest doubles in value in 7.0 years. At the same interest rate, how long would it take the value of the account to triple? (A) 4.4 years (B) 9.8 years (C) 10.5 years (D) 11.1 years (E) 21.0 years
Problem 50
In Exercises \(47-50,\) use integration by parts to establish the reduction formula. $$\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x$$
Problem 50
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int 4 \cos ^{2} x d x, \quad \cos 2 x=1-2 \cos ^{2} x$$
Problem 50
Multiple Choice A sample of Ce-143 (an isotope of cerium) loses 99\(\%\) of its radioactive matter in 199 hours. What is the half-life of \(\operatorname{Ce}-143 ?\) (A) 4 hours (B) 6 hours (C) 30 hours (D) 100.5 hours (E) 143 hours
Problem 51
Multiple Choice In which of the following models is \(d y / d t\) directly proportional to \(y\) ? $$\begin{aligned} \text { I. } y &=e^{k t}+C \\ \text { II. } & y=C e^{k t} \\\ \text { III. } & y=28^{k t} \end{aligned}$$ (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III
