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1. Show that -∞∫∞ (1 + x) / (1 + x2) dx diverges, but limt→∞-t∫t (1 + x) / (1 + x2) dx = π.
2. Use the Comparison Theorem to decide if these integrals converge or diverge.
a. 1∫∞ (4 + e-x) / x dxb. 0∫∞ arctan x / (2 + ex) dx
3. The Gamma function is defined by Γ(t) = 0∫∞ xt-1 e-x dx. Use integration by parts to prove that Γ(z + 1) = z Γ(z).
4. Consider the curve y = 1/x for x ≥ 1.
a. Show that the area under the curve is infinite.
b. Show that the volume of the solid of revolution enclosed by the curve is finite, and compute it. [This solid of revolution is known as Gabriel's Horn.]
a person flying a kite hold a string 4 feet above ground level. the string of the kite is taut and makes an angle of
Given three points, there is one line that can be drawn through them if the points are colinear. If the three points are noncolinear,there are three lines that can be drawn through pairs of points.
What is the probability that the tire you recently purchased will last at most 81,000 km.?
How many edges does an n-cube have? Give your reason. How many (main) diagonals does a 4-dim cube have? Are any two of them perpendicular to each other?
Find the probability that during any given day three machines will break down.
An entrepreneur has the following cost and revenue functions: TC = 30 + 3Q and TR = 4Q. calculate the equilibrium quantity algebraically and graphically at the break-even point.
Using the various properties Closure, Commutative etc. and Distributive, explain how you would compute the following, using quick and easy methods of association such as in 7+3 =10, 5'9=10 etc. and show the steps of your process.
consider the following data values of a variable x and yx 54369810y 78105231construct a scatter diagram for the data
There are six sections of Math with a total enrollment of one hundred and ninety nine students, what is the smallest possible number of students in the section with the largest enrollment?
Let |G| be finite and N be a normal subgroup of G. If xN is an element of G/N and has order a power of p, show that there exists a y element of G such that |y| is a power of p and yN=xN.
A pool measuring 8 meters by 22 meter is surrounded by a path of uniform width. If the area of the pool and the path combined is 392 square meters, what is thw width of the path.
How far could each animal move in an hour A sanil can move 40 feet in 1/4 an hour a turtle can move 300 feet in an hour a sloth can move 50 feet in 1/3 an hour
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