How to evaluate the definite integral of 1/(1+square root x), for limis of integration 0 and 4?

First, we'll determine the indefinite integral of the
function y. For this reason, we'll replace the variable x by
t^2.

x = t^2

We'll
differentiate both sides:

dx =
2tdt

Int  dx/(sqrt x + 1) = Int [1/(t +1)]*(2t
dt)

Int [1/(t +1)]*(2t dt) = 2Int (t+1)dt/(t +1) - 2Int
dt/(t+1)

Int [1/(t +1)]*(2t dt) = 2Int dt - 2Int
dt/(t+1)

We'll calculate the 1st
term:

2Int dt = 2t + C

We'll
calculate the 2nd term:

2Int dt/(t+1) = 2 ln|t+1| +
C

Int  dx/(sqrt x + 1) = 2sqrt x - 2 ln |sqrt x + 1| +
C

We'll apply Leibniz Newton to determine the definite
integral:

Int  dx/(sqrt x + 1) = F(4) -
F(0)

F(4) = 2sqrt 4 - 2 ln |sqrt 4 +
1|

F(4) = 4 - 2 ln 3

F(4) = 4
- ln 9

F(0) = 2sqrt 0 - 2 ln |sqrt 0+
1|

F(0) = 0 - ln 1

F(0) =
0

The definite integral of the given
function, if the limits of integration are x = 0 to x = 4,
is: Int dx/(sqrt x + 1) = 4 - ln
9

Summarize the poem "Loveliest of Trees" by A. E. Housman.

A, E. Housman writes about making the most of the moment
in his poem “The Loveliest of Trees.” No one will live forever, so squeeze every
possible minute of the enjoyment of nature.

The setting of
the poem is Easter, and spring has brought the lovely cherry trees to bloom.  The poet
is riding in a carriage or on horseback  in a rural area. The narration is first person
with the poet as the
speaker.

Summary

The
first stanza describes a beautiful scene.  The cherry trees are so covered with blossoms
that the branches hang down from their bounty. 

readability="13">

Loveliest of trees, the cherry
now

Is hung with bloom along the
bough,

And stands about the woodland
ride

Wearing white for
Eastertide.

The trees stand
along the way as the speaker goes for a ride.  It is the perfect setting for the Easter
time with the trees wearing their white.  The speaker implies a connection to the
religious time and the white of the trees symbolizing the purity and innocence of
Christ.

In the second stanza, we learn that the speaker is
a little more than twenty years old. Of his "threescore years and ten" (seventy
years--the life expectancy of man laid out in Psalms 90 of the Bible), he will only have
fifty more years to enjoy the beauty of the earth.

The
third stanza explains that fifty years is really not enough time to relish the beauty of
the earth, the spring, and the lovely cherry trees.  Using a clever idea, he will be
able to observe the cherry trees in winter as well as spring. In winter, the trees will
be covered in white, but this will be snow; in spring, they are covered with the
luscious white blooms. He will travel again in the woodland area to take in the beauty
of the cherry trees.

The poet conveys the idea that spring
does not have the monopoly on beauty for the seasons.  Each season has its own kind of
glory. The fall has the leaves; the winter its snow; the summer its flowers; and the
spring its green---all are a part of the landscape which in itself is a work of
art.

Life is short, and travels rapidly by. The only thing
a person can do is relish each moment that he has.  Be excited about life and its
beauty.  Nature will go on whether one enjoys it or not.  Nature is indifferent to man
and does not wait on man to enjoy its loveliness.  The speaker will appreciate the
spring and the exquisite trees as long as he walks the earth. 

Compare and contrast Keats' "La Belle Dame sans Merci" with Tennyson's "The Lady of Shalott"?

These two poems Keats' "La Belle Dame sans Merci" and
Tennyson's "The Lady of Shalott" are expressions of unrequited love, the contrasts
between the world of reality and the world of imagination, and the difficulty of the two
worlds coexisting.

Both poems are about unrequited love.
Keats' poem concerns a pale knight who after experiencing a tryst with a beautiful
"fairy's child" feels abandoned and dejected on a "cold hillside."  In Tennyson's poem,
the lady leaves her secluded shadowy world to find enter the real world that Lancelot
inhabits.  She dies before she reaches Camelot and meets
Lancelot.

Both involve the contrasting worlds of
imagination and reality.  Keats' knight after experiencing the extraordinary world of
the imagination cannot bear the dullness of the real world; Tennyson's lady, after only
knowing the shadowy secluded world of fantasies, cannot survive in the bright world of
reality.

How does Shakespeare use language to create sympathy for Othello?I'm looking for quotes mainly, that i can analyse

The agony that Othello endures when he feels that he has
lost Desdemona is expressed so eloquently and so passionately that it is difficult not
to feel sympathy for Othello even when the reader knows that he is wrong.  Two speeches
are particularly poignant.  The first occurs in Act 3 after Iago's words have begun to
create suspicion and jealousy:

readability="19">

I had been happy if the general
camp,

Pioners and all, had tasted her sweet
body,

So I had nothing know.  Oh, now
forever

Farewell the tranquil mind!  Farewell
content!

Farewell the plumed troop and the big
wars

That make ambition virtue!  Oh farewell . .
.

In this speech Othello's
war and soldier references equate Desdemona's love to all that he feels is important in
this world, to all that has defined him as a man. Everything that has made life seem
worthwhile is lost to him now that he doubts Desdemona's love for
him.

Later, Othello's anguish over the ocular proof Iago
has provided of Desdemona's infidelity also reveal his deep emotions.  He wavers between
his anger toward Desdemona and his admiration for her in such lines
as

Hang her! I
do but say what she is, so delicate with her needle, an admirable musician--oh, she will
sing the savageness out of a bear--of so high and plenteous wit and
invention--

Here, we see
Othello expressing within the same sentences his struggle between his love for Desdemona
and his belief that she should be punished for her
indiscretions.

And of course, Othello's unmitigated remorse
for killing Desdemona and his refusal to rationalize his actions also create sympathy
for this noble general:

readability="14">

Cold, cold, my
girl!

Even like thy chastity. Oh, cursed
slave!

Whip me, ye
devils,

From the possession of this heavenly
sight!

Blow me about in winds! Roast me in
sulphur!

Wash me in steep-down gulfs of liquid
fire!

O Desdemona! Dessdemona!
Dead!

Oh! Oh!
Oh!

The heaven and hell
imagery here shows Othello's belief that his actions have damned him and that
Desdemona's innocence will secure her a place in heaven. The short exclamatory lines
indicate his extreme emotion, and the hyperboles show that he  fully understands  that
he will suffer not only in this life but in the
afterlife.

And finally, Othello's last speech, which places
the blame only on himself for throwing "the dearest pearl away" is the last time that
treasure imagery is used to describe Desdemona, and shows once more Othello's integrity
for admitting his wrongdoing and his willingness to suffer the consequences for his
actions.

In this way, pathos for Othello is
established.

In the anonymous Old English poem Beowulf, how does Beowulf feel his life was saved? (See lines 1651-1887 of the Seamus Heaney translation)

In lines 1651-1887 of the Seamus Heaney translation of the
Old English epic poem Beowulf, Beowulf offers several reasons why
his life was saved in his battle with Grendel’s mother.  His description of his fight
with her includes the following elements:

• He
  acknowledges that the battle was

readability="11">

. . . hard-fought, a desperate
affair

that could have gone badly; if God had not helped
me,

the outcome would have been quick and fatal.
(1656-58)

He thus
acknowledges his utter dependence on God and gives God the glory for the
victory.

• He acknowledges the insufficiency of
  his own strength and weaponry and thus credits God with his victory
  (1659-1661).


• He gives God credit for allowing him to spot
  a useful weapon (1661-64).


• He indicates that, having
  received help from God, he then conducted himself wisely and bravely, although he
  doesn’t boast (1665-66). He thus shows that God helps those who try to help themselves,
  especially if their causes is virtuous and just.


• He
  offers his victory as a gift to Hrothgar and the Danes
  (1671-76).


• He shows respect to Hrothgar by giving
  Hrothgar what is left of the weapon used to slay Grendel’s mother (1677-79).
  

In short, Beowulf speaks precisely as an ideal
early Christian warrior should speak: he glorifies God; he shows respect to those above
him on the “great chain of being”; he is modest; he is brave, he is eloquent; he is
wise; he acknowledges the power of earthly evil while also paying tribute to the greater
power of God; he sees himself as an instrument of God’s providence; and he conducts
himself with dignity, humility, sincerity, and prudence.

How do I insert a blue line to the left of the quoted text just like your editors do it. It is beautiful and professional looking.

What you are referring to is a block quote. It is often
used to show that the text which has been marked is not a creation of the editor but has
been taken from a source and used verbatim. These are of great help to explain things in
a better way when a question is being answered.

When a
question is being asked, there is no necessity for this to be done and therefore no
option has been provided for text to be marked as you have described. If you would like
to mark anything as a direct quote, you could just enclose it within double quotes and
place it as a separate paragraph.

Even if you are not an
editor you can provide helpful answers to others on this site. When answering a
question, you can mark a direct quote by highlighting it and using the double quote
symbol below the text box.

Find the x-intercept and y-intercept for f(x) f(x)=x^2-3x+1

To determine x intercept, we'll put y = f(x) =
0.

But f(x) = x^2-3x+1 => x^2-3x+1 =
0

We'll apply quadratic
formula:

x1 = [3+sqrt(9 -
4)]/2

x1 = (3+sqrt5)/2

x2 =
(3-sqrt5)/2

The x intercepts are: ((3+sqrt5)/2 , 0) and
((3-sqrt5)/2) , 0).

To determine y intercepts, we'll cancel
the x coordinate.

f(0) = 0^2 - 3*0 +
1

f(0) = 1

The y intercept is
(0 , 1).

The x intercepts are ((3+sqrt5)/2 ,
0) and ((3-sqrt5)/2) , 0) and the y intercept is (0 ,
1).