Definite Integrals

[Azerail]

I have two questions that I really don't have a clue how to answer. As such, I can't show any work because I have no idea where to even begin. Any help would be appreciated.

1. Evaluate limit of n approaching infinity. Express answer as a reduced fraction in the form a/b.

81/n^4[n^2(n+1)^2/4]

2. Which of the following are true?
a.∫b/a f(x)dx = ∫a/b f(x)dx
b.∫b/a kf(x)dx = ∫b/a f(kx)dx
c.∫a/a f(x)dx=0
d.∫b/a (f(x)+g(x))dx=∫a/b f(x)dx-∫b/a g(x)dx
e.∫b/a f(x)dx+∫c/b f(x)dx=∫c/a f(x)dx

 

[Ishuda]

I have two questions that I really don't have a clue how to answer. As such, I can't show any work because I have no idea where to even begin. Any help would be appreciated.

1. Evaluate limit of n approaching infinity. Express answer as a reduced fraction in the form a/b.

81/n^4[n^2(n+1)^2/4]

2. Which of the following are true?
a.∫b/a f(x)dx = ∫a/b f(x)dx
b.∫b/a kf(x)dx = ∫b/a f(kx)dx
c.∫a/a f(x)dx=0
d.∫b/a (f(x)+g(x))dx=∫a/b f(x)dx-∫b/a g(x)dx
e.∫b/a f(x)dx+∫c/b f(x)dx=∫c/a f(x)dx

For (1), the way it is written I think it would be interpreted as
\(\displaystyle \frac{81}{n^4}\, \frac{n^2\, (n+1)^2}{4}\)
If that is the case, I would re-write it as
\(\displaystyle \frac{81}{4}\, \frac{n^2\, (n+1)^2}{n^4}\, =\, \frac{81}{4}\, (\frac{n+1}{n})^2\)

For (2), the way the problems are written, I would assume the b/a is used to express upper and lower limits [upper/lower], for example e would become
e. \(\displaystyle \underset{a}{\overset{b}{\int}}\, f(x)\, dx\, +\, \underset{b}{\overset{c}{\int}}\, f(x)\, dx\, =\, \underset{a}{\overset{c}{\int}}\, f(x)\, dx\)

What are your thoughts? What have you done so far? Please show us your work so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

 

[Steven G]

I have two questions that I really don't have a clue how to answer. As such, I can't show any work because I have no idea where to even begin. Any help would be appreciated.

1. Evaluate limit of n approaching infinity. Express answer as a reduced fraction in the form a/b.

81/n^4[n^2(n+1)^2/4]

2. Which of the following are true?
a.∫b/a f(x)dx = ∫a/b f(x)dx
b.∫b/a kf(x)dx = ∫b/a f(kx)dx
c.∫a/a f(x)dx=0
d.∫b/a (f(x)+g(x))dx=∫a/b f(x)dx-∫b/a g(x)dx
e.∫b/a f(x)dx+∫c/b f(x)dx=∫c/a f(x)dx

I am assuming that you meant to write [81/n^4][n^2(n+1)^2/4]

Here is the general rule: If lim as x goes to infinity of f(x)/g(x) then

if deg f(x) > deg g(x), then the lim is +/- infinity. To get the sign (+ or -) use the sign from a/b where a is the leading coefficient of f(x) and b is the leading coefficient of g(x).

if deg f(x) < deg g(x), then the lim is 0.

if deg f(x) = deg g(x), then the lim equals a/b

a.∫b/a f(x)dx = ∫a/b f(x)dx Let's suppose ∫f(x)dx=F(x). Then ∫b/a f(x)dx =F(b)-F(a) while ∫a/b f(x)dx= F(a)-F(b)
b.∫b/a kf(x)dx = ∫b/a f(kx)dx kf(x) generally does not equal f(kx)
c.∫a/a f(x)dx=0 ∫a/a f(x)dx = F(a)-F(a)
d.∫b/a (f(x)+g(x))dx [F(b)+G(b)]-[F(a)+G(a)] while ∫a/b f(x)dx-∫b/a g(x)dx [F(a)-F(b)]-[G(b)-G(a)]
e.∫b/a f(x)dx+∫c/b f(x)dx= [F(b)-F(a)]+[F(c)-F(b)] while ∫c/a f(x)dx =F(c)-F(a)
For the 2nd problem all I did is use the definition. So which ones do you think are valid?