Chapter 6: Problem 39
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{x \ln x}$$
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Multiple Choice If \(\int x^{2} \cos x d x=h(x)-\int 2 x \sin x d x,\) then \(h(x)=\) (A) \(2 \sin x+2 x \cos x+C\) (B) \(x^{2} \sin x+C\) (C) \(2 x \cos x-x^{2} \sin x+C\) (D) \(4 \cos x-2 x \sin x+C\) (E) \(\left(2-x^{2}\right) \cos x-4 \sin x+C\)
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{2}^{5} \frac{d x}{2 x-3}$$
Multiple choice \(\int x \csc ^{2} x d x=\) (A) \(\frac{x^{2} \csc ^{3} x}{6}+C\) (B) \(x \cot x-\ln |\sin x|+C\) (C) \(-x \cot x+\ln |\sin x|+C\) (D) \(-x \cot x-\ln |\sin x|+C\) (E) \(-x \sec ^{2} x-\tan x+C\)
Consider the integral \(\int x^{n} e^{x} d x .\) Use integration by parts to evaluate the integral if (a) \(n=1\) (b) \(n=2\) (c) \(n=3\) (d) Conjecture the value of the integral for any positive integer \(n\) (e) Writing to Learn Give a convincing argument that your conjecture in part (d) is true.
Solving Differential Equations Let \(\frac{d y}{d x}=\frac{1}{x}\) . (a) Show that \(y=\ln x+C\) is a solution to the differential equation in the interval \((0, \infty)\) (b) Show that \(y=\ln (-x)+C\) is a solution to the differential equation in the interval \((-\infty, 0)\) (c) Writing to Learn Explain why \(y=\ln |x|+C\) is a solution to the differential equation in the domain \((-\infty, 0) \cup(0, \infty)\) (d) Show that the function \(y=\left\\{\begin{array}{l}{\ln (-x)+C_{1}} \\ {\ln x+C_{2}}\end{array}\right.\) \(x<0\) \(x>0\) is a solution to the differential equation for any values of \(C_{1}\) and \(C_{2}\)
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