I am a mathematics PhD student. I really enjoy helping students learn calculus. My research interests lie in the field of topology. Outside of math, I enjoy running and learning philosophy.
$1-80$ Evaluate the integral.$$\int_{0}^{\pi / 4} \tan ^{5} \theta \sec ^{3} \theta d \theta$$
$1-80$ Evaluate the integral.$$\int \frac{d x}{x+x \sqrt{x}}$$
$1-80$ Evaluate the integral.$$\int \frac{x \ln x}{\sqrt{x^{2}-1}} d x$$
$1-80$ Evaluate the integral.$$\int \sqrt{x} e^{\sqrt{x}} d x$$
$1-80$ Evaluate the integral.$$\int \frac{1}{x+\sqrt[3]{x}} d x$$
$1-80$ Evaluate the integral.$$\int \frac{\sqrt{x}}{1+x^{3}} d x$$
Write the converse and inverse of the statement. If it isspring, then some people go hiking. converse:_________________________________________________________________________inverse:____________________________________________________________________
2) Predict the next term in the sequence -4, -1, 14, 47, 104, 191, 314, ?
Place numbers in blanks so that the following sentence is true:"In this sentence, the number of occurrences of 0 is _, of 1 is _,of 2 is _, of 3 is_, of 4 is _, of 5 is _, of 6 is _, of 7 is _, of8 is _, and of 9 is _."(Problem Solving course; all the information for what is given,is stated above already).
Theorem: Every bounded, monotonic sequence is convergent.
A) Describe what this means (you may draw a picture also.)
B) Give an example of a bounded monotonic sequence and then show it converges.
C) Given an example of a bounded sequence that is not monotonic and does not converge. You must argue your point.
For each of the following degree sequences, either drawa simple graph with the degree sequence,or explain why no such graph exists.a) Six vertices with degreesequence (5,4,3,2,1,0)(5,4,3,2,1,0)b) Six vertices with degree sequence (5,4,3,3,3,2)
(a) Find the Maclaurin series for cos(x^2) using the Maclaurin series for cos x. (Your answer should use summation notation.)(b) Express the integral ∫₀¹ cos(x^2) dx as an infinite series, using part (a). (Your answer should use summation notation.)
