# Consider the integral as attached, To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals for the given integral to converge? int_0^3 10/(x^2-2x)dx.

Question
Applications of integrals
Consider the integral as attached, To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals for the given integral to converge?
$$\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}$$.

2021-03-08
Step 1
Consider the integral $$\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}$$.
To determine the convergence or divergence of the integral,and we need to analyze how many improper integrals are there.
and what must be true of each of these integrals for the given integral to converge.
Step 2
First we will see the dicontinuities of the function $$\displaystyle\frac{{10}}{{{x}^{{2}}-{2}{x}}}$$
$$\displaystyle{x}^{{2}}-{2}{x}={0}\ge{x}={0}{\quad\text{or}\quad}{x}={2}$$
We can write
$$\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}=\lim_{{{a}\rightarrow{0}^{+}}}{\int_{{a}}^{{b}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}{\left.{d}{x}\right.}+\lim_{{{c}\rightarrow{2}^{{-}}}}{\int_{{b}}^{{c}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}+\lim_{{{d}\rightarrow{2}^{{-}}}}{\int_{{d}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}$$
$$\displaystyle{L}{e}{t}{c}\in{\left({0},{2}\right)}$$
The integral must split in three improper integrals to contain one limit per integral. As 0 is the left integral limit we only need the right hand limit.
The dicontinuity at x=2 lies inside the interval (0,3) so we will consider both the limits.
Step 3
Now the limit must exists for all the three integrals to be convergent.
and each of the three integrals must be convergent for the integral $$\displaystyle{\int_{{0}}^{{3}}}\frac{{10}}{{{x}^{{2}}-{2}{x}}}$$ dx to be convergent.

### Relevant Questions

The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
$$\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}$$
where A is the cross-sectional area of the vehicle and $$\displaystyle{C}_{{d}}$$ is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity $$\displaystyle\vec{{{v}}}$$. Find the power dissipated by form drag.
Express your answer in terms of $$\displaystyle{C}_{{d}},{A},$$ and speed v.
Part B:
A certain car has an engine that provides a maximum power $$\displaystyle{P}_{{0}}$$. Suppose that the maximum speed of thee car, $$\displaystyle{v}_{{0}}$$, is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power $$\displaystyle{P}_{{1}}$$ is 10 percent greater than the original power ($$\displaystyle{P}_{{1}}={110}\%{P}_{{0}}$$).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, $$\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}$$, is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
Factor each polynomial. If a polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary:
$$\displaystyle{60}{m}^{{{4}}}-{120}{m}^{{{3}}}{n}+{50}{m}^{{{2}}}{n}^{{{2}}}$$
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Factor each polynomial. If a polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary:
$$\displaystyle{x}^{{{2}}}+{4}{x}-{5}$$
Use a table of integrals to evaluate the definite integral.
$$\displaystyle{\int_{{0}}^{{3}}}\sqrt{{{x}^{{2}}+{16}}}{\left.{d}{x}\right.}$$
Factor each polynomial. If a polynomial cannot be factored, write prime. Factor out the greatest common factor as necessary:
$$\displaystyle{3}{a}^{{{2}}}+{10}{a}+{7}$$
$$\displaystyle{15}{y}^{{{2}}}+{y}-{2}$$
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}-{3}{x}^{{{2}}}-{2}{x}+{3}$$
$$\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}$$
$$\displaystyle{3}{m}^{{{3}}}+{12}{m}^{{{2}}}+{9}{m}$$