Problem 74
Evaluate the following integrals. $$\int \frac{e^{3 x}}{e^{x}-1} d x$$
Problem 74
Integrating inverse functions Assume \(f\) has an inverse on its domain. a. Let \(y=f^{-1}(x)\) and show that $$\int f^{-1}(x) d x=\int y f^{\prime}(y) d y$$. b. Use part (a) to show that $$\int f^{-1}(x) d x=y f(y)-\int f(y) d y$$. c. Use the result of part (b) to evaluate \(\int \ln x \, d x\) (express the result in terms of \(x\) ). d. Use the result of part (b) to evaluate \(\int \sin ^{-1} x d x\). e. Use the result of part (b) to evaluate \(\int \tan ^{-1} x d x\).
Problem 74
Finding constants with a computer algebra system Give the -appropriate form of the partial fraction decomposition of the expression, and then use a computer algebra system to find the unknown constants. $$\frac{3 x^{2}+2 x+1}{(x+1)^{3}\left(x^{2}+x+1\right)^{2}}$$
Problem 75
Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function \(s(t)=e^{-t} \sin t\) a. Graph the position function. At what times does the oscillator pass through the position \(s=0 ?\) b. Find the average value of the position on the interval \([0, \pi]\) c. Generalize part (b) and find the average value of the position on the interval \([n \pi,(n+1) \pi],\) for \(n=0,1,2, \ldots\) d. Let \(a_{n}\) be the absolute value of the average position on the interval \([n \pi,(n+1) \pi],\) for \(n=0,1,2, \ldots . .\) Describe the pattern in the numbers \(a_{0}, a_{1}, a_{2}, \dots\)
Problem 75
Volume of a solid Consider the region \(R\) bounded by the graph of \(f(x)=\sqrt{x^{2}+1}\) and the \(x\) -axis on the interval \([0,2] .\) Find the volume of the solid formed when \(R\) is revolved about the \(y\) -axis.
Problem 75
Exact Simpson's Rule a. Use Simpson's Rule to approximate \(\int_{0}^{4} x^{3} d x\) using two subintervals \((n=2) ;\) compare the approximation to the value of the integral. b. Use Simpson's Rule to approximate \(\int_{0}^{4} x^{3} d x\) using four subintervals \((n=4) ;\) compare the approximation to the value of the integral. c. Use the error bound associated with Simpson's Rule given in Theorem 8.1 to explain why the approximations in parts (a) and (b) give the exact value of the integral. d. Use Theorem 8.1 to explain why a Simpson's Rule approximation using any (even) number of subintervals gives the exact value of \(\int_{a}^{b} f(x) d x,\) where \(f(x)\) is a polynomial of degree 3 or less.
Problem 75
Exploring powers of sine and cosine a. Graph the functions \(f_{1}(x)=\sin ^{2} x\) and \(f_{2}(x)=\sin ^{2} 2 x\) on the interval \([0, \pi] .\) Find the area under these curves on \([0, \pi]\) b. Graph a few more of the functions \(f_{n}(x)=\sin ^{2} n x\) on the interval \([0, \pi]\), where \(n\) is a positive integer. Find the area under these curves on \([0, \pi] .\) Comment on your observations. c. Prove that \(\int_{0}^{\pi} \sin ^{2} n x d x\) has the same value for all positive integers \(n\) d. Does the conclusion of part (c) hold if sine is replaced by cosine?
Problem 75
Evaluate the following integrals. $$\int_{1}^{3} \frac{\tan ^{-1} \sqrt{x}}{x^{1 / 2}+x^{3 / 2}} d x$$
Problem 75
Finding constants with a computer algebra system Give the -appropriate form of the partial fraction decomposition of the expression, and then use a computer algebra system to find the unknown constants. $$\frac{x^{4}+3 x^{2}+1}{x\left(x^{2}+1\right)^{2}\left(x^{2}+x+4\right)^{2}}$$
Problem 75
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. It is possible for a computer algebra system to give the result \(\int \frac{d x}{x(x-1)}=\ln (x-1)-\ln x\) and a table of integrals to give the result \(\int \frac{d x}{x(x-1)}=\ln \left|\frac{x-1}{x}\right|+C\) b. A computer algebra system working in symbolic mode could give the result \(\int_{0}^{1} x^{8} d x=\frac{1}{9},\) and a computer algebra system working in approximate (numerical) mode could give the result \(\int_{0}^{1} x^{8} d x=0.11111111\).
