Problem 46
\(\int \csc x d x \quad(\)Hint\(:\) Multiply the integrand by \(\frac{\csc x+\cot x}{\csc x+\cot x}\) and then use a substitution to integrate the result.)
Problem 46
In Exercises \(45-48\) , use Euler's Method with increments of \(\Delta x=-0.1\) to approximate the value of \(y\) when \(x=1.7\) \(\frac{d y}{d x}=1+y\) and \(y=0\) when \(x=2\)
Problem 46
Rules of 70 and 72 The rules state that it takes about 70\(/ i\) or 72\(/ i\) years for money to double at \(i\) percent, compounded continuously, using whichever of 70 or 72 is easier to divide by \(i .\) (a) Show that it takes \(t=(\ln 2) / r\) years for money to double if it is invested at annual interest rate \(r\) (in decimal form) compounded continuously. (b) Graph the functions $$y_{1}=\frac{\ln 2}{r}, \quad y_{2}=\frac{70}{100 r}, \quad\( and \)\quad y_{3}=\frac{72}{100 r}$$ in the \([0,0.1]\) by \([0,100]\) viewing window. (c) Writing to Learn Explain why these two rules of thumb for mental computation are reasonable. (d) Use the rules to estimate how long it takes to double money at 5\(\%\) compounded continuously. (e) Invent a rule for estimating the number of years needed to triple your money.
Problem 47
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int \sin ^{3} 2 x d x, \quad \sin ^{2} 2 x=1-\cos ^{2} 2 x$$
Problem 47
In Exercises \(45-48\) , use Euler's Method with increments of \(\Delta x=-0.1\) to approximate the value of \(y\) when \(x=1.7\) \(\frac{d y}{d x}=x-y\) and \(y=2\) when \(x=2\)
Problem 47
Partial Fractions with Repeated Linear Factors If \(f(x)=\frac{P(x)}{(x-r)^{m}}\) is a rational function with the degree of \(P\) less than \(m,\) then the partial fraction decomposition of \(f\) is \(f(x)=\frac{A_{1}}{x-r}+\frac{A_{2}}{(x-r)^{2}}+\ldots+\frac{A_{m}}{(x-r)^{m}}\) For example, \(\frac{4 x}{(x-2)^{2}}=\frac{4}{x-2}+\frac{8}{(x-2)^{2}}\) Use partial fractions to find the following integrals: (a) \(\int \frac{5 x}{(x+3)^{2}} d x\) (b) \(\int \frac{5 x}{(x+3)^{3}} d x \quad(\) Hint: Use part (a).)
Problem 47
True or False If \(d y / d x=k y,\) then \(y=e^{k x}+C .\) Justify your answer.
Problem 47
In Exercises \(47-50,\) use integration by parts to establish the reduction formula. $$\int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x$$
Problem 48
True or False The general solution to \(d y / d t=2 y\) can be written in the form \(y=C\left(3^{k t}\right)\) for sor some constants \(C\) and \(k .\) Justify your answer.
Problem 48
In Exercises \(47-52,\) use the given trigonometric identity to set up a \(u\) -substitution and then evaluate the indefinite integral. $$\int \sec ^{4} x d x, \quad \sec ^{2} x=1+\tan ^{2} x$$
