chengyewang/TexOCR-eval huggingface dataset:
Dear Friends:-

Yours awaited me on my return from Fairfax Tuesday evening (or night rather). I left here on Sunday morning in the rain in company with Mr. Welles, Mrs. Morell, Mr. Haskell, Mr. Alvord, et al., and took train at Morgan Rulley's office, and soon found ourselves at Fairfax.

I cannot tell you the scenes which awaited our eyes; the wounded were constantly coming; but no hospitals this time; only God's great one under the blue canopy. The men were brought down from the field and laid on the ground beside the train, and so back up the hill till they covered acres. The bales of hay for forage were broken open and the ground was ``littered'' like ``bedding'' for horses. They came till dark and then it was dark indeed: one lantern on the ground; no use requisition for candles; drew a few. The wind blew just enough to put them out every few minutes, and the men lay so thick we could not take one step in the dark. By midnight there must have been three thousand helpless men lying in that hay. We had two water buckets---five dippers---the stores which we carried to eat besides hard crackers. My one stove pan which I remembered to take, and this made coffee for them. All night we made compresses and slings and bound up and wet wounds when we could get water, for what we could travel miles in the dark over those poor helpless wretches, in terror lest some one candle fall into the hay and consume them all. At length morning came and we sent up the train with 1250. Next 1000, next 1100, next 940 and so on. Still the ambulances came down and the cars went out and we worked on. Took the meat from our own sandwiches and gave it to them, and broke the bread into wine and water to feed the poor sinking wretches as they lay in the ambulances or on the ground.

On Monday the cavalry appeared in the wood opposite and a raid was hourly expected. I neglected to tell you that Mrs. Fales sent to me before I started to know if she would go with me and I had the train wait and sent back an ambulance for her and her stores and this made three ladies---Mrs. Carter is away.

On Monday p.m. all the wounded then in were sent off and the danger became so imminent that Mrs. Fales thought it best to leave, as those only ``went for strong''. I begged to be excused from accompanying her as the ambulances were up to the field for more wounded and I knew I should never leave a wounded man there---if I knew it, though I were taken prisoner forty times.
At six o'clock it commenced to thunder and lighten, and all at once the artillery began to play joined by the musketry, about two miles distant. We sat down in our tent and waited to see them break in upon us; but Reno's forces held them back. The old 21st Mass. lay between us and the enemy and they couldn't pass.

God only knows who is lost, I do not; for the next day all fell back. Poor Kearney, Stephens and Webster were brought in, and in the p.m. Kearney and Heintzelman's divisions fell back through our camp on their way to Alexandria. I know this was the last. We put the thousand wounded we had then into the train. I took one car load of them, Mrs. Morell and others. The men took to horses. We steamed off and two hours later there was no Fairfax Station. Reached Alexandria at ten o'clock at night, and oh, the reproach which met those poor men at the train. The people of the Island are the most noble band I ever saw or heard of. I stood in my car and fed the men 'till they could eat no more; then the people would take us home and feed us and after this we came home.

I had slept 1-1/4 hours since Saturday night; but I am well and strong and wait to go again if I am needed. Our forces are all back again in the old places around the city. McClellan's army here again and he in command of it all.

I am going to reach for my friends now. I have told you nothing of the old friends who met us among the wounded and dying on that bloody field. I have no heart to tell it today. Two huge boxes from Jersey have arrived. I don't know where we shall need them next. To Archie and Lizzie Shaver.

Yours,

Clara Barton.

Frankfort, N.Y

Miss Clara H. Barton,

I have conferred with Gen. Wadsworth on the subject of obtaining a horse for your use and he has directed that a horse be placed at your disposal as soon as a suitable one can be found.

Very respectfully

Yr obt. servt.

John P. Sherburne

Asst. Adj. Gen.
Dear friends --- Mr Wells returns tomorrow, and I improve the opportunity to send a line by him to you, not feeling quite certain if posted matter reaches directly when sent from the Army. We reached Acquia Creek safely in the time anticipated, and to my great joy I learned immediately that our old friends Capt (Maj.) Head (of the 21st) was D. Master; as soon as the Boat was unloaded he came on board and spent the remainder of the evening with me --- we had a home chat I assure you --- remained till the next day, sent a bbl of apples etc up to the Capts quarters, and proceeded with the remainders of our luggage for which it is needless to say ready transportation.
was found and the Capt. chided me for having left any thing behind at the depot—as I told him I had done,—On reaching Falmouth Station we found another old friend, Capt. Bailey in charge who instituted himself as watch over the goods until he sent them all up to Head Quarters—My Ambulance came through that PM, but for fear it might not. Col Sterges had his taken down for me, and had supper arranged and a splendid serenade: I dont know how we could have had a warmer ``welcome home'' as the Officers termed it.

Head Quarters are in the door yard of a farm house, one room of which is occupied by Miss B, and myself. My wagons are a little way from me, out of sight, and I am wishing for a tent and stove to pitch and live near them. The weather is cold, and the ground covered with snow, but I could make me comfortable with a good tent, floor and stove, and should prefer it to a room in a rebel house one so generally occupied—

The 2d are a few rods from me,—many of the officers call to see me every day. Col Clark is very neighborly, he is looking finely now he was in this P.O., and was going in search of Col Morse when he thought to be a mile or two distant; I learned to night that the 15th are only some three miles away, the 36th I cannot find yet. I have received horses for them and shall get on track soon I trust. Of living movements nothing can be said with certainty, no two persons, not even the Gen'ls agree in reference to the future programme, the snow appears to have deranged the plans very seriously. I have rec'd calls from two Genl's ladies and in the course of conversation I discovered that their views were entirely different. Genl Burnside stood a long time in front of my door today but to my astonishment, he did not express his opinion; Strange!

I have not suffered for want of the boots yet, but should find them convenient I presume, and shall be glad to see them.—The sore finger is much the same, not very troublesome although somewhat so
me at home without a question under the circumstances he was wise. Or perhaps he thought I should come go without the aid or help of the Expedition if I could be of any service all of which are surmises. I made the request of the Gen. of that particular time because the service of the petitioners made it necessary for me to say something—and introduce when I should be at liberty to ask of such friends as I had in the Regiment if I could serve them in any way when an officer more suffering than I was and deserve should come upon them more readily than armed men—and this is what I have to say to you—and my dear friend Mr Ives—That if in the course of your service in the Western soil and climate there be a time \& place when my strength, firm health, and willing hand and somewhat of experience can benefit the great cause holy cause in which it is your pleasure privilege to serve—permit me to forego comfort and quiet life, your wife like mine and let me come and do it—I honor the thousands who are now ready and willing waiting, and I offer over—because I can do better as sole teacher no other, but I am stronger, better acclimated, have home, health, and better able to forego comforts than ladies in general. I have already many men, and what is more my life if I should lose it is worth less to others; there are few to mourn; and among the most willing, your fellow servant I

Nothing less than a strong conviction of duty, owed to one of the brave defenders of our Nation's honor would induce me to intrude for a moment upon the already burdened, and limited term of action yet remaining to your Honorable Body. During the late Battle of Fredericksburg the 21st Massachusetts Regiment of Volunteers were ordered to charge upon a Battery across an open field: in the terrible fire which assailed them the orders were three times in quick succession before of their support; the third time they were led by Sergeant Thomas Plunkett of Co. E, and borne over some three hundred yards of open space, when a shell from the enemies Battery in its murderous course killed three men of the regiment—and shattered both arms
of the Serjeant. He could no longer support the colors upright; but planting his foot against the Staff, he endeavoured to hold them up, while he strove by his shouts amid the confusion to attract attention to their condition; for some minutes he sustained them against his right arm, torn and shattered just below the shoulder, while the blood poured over and among the sacred folds, literally obliterating the stripes, leaving as fit emblem of such heroic sacrifice, only the crimson and the stars. Thus drenched in blood, and sensibly the fury of eight battles, the noble Standard could be no longer borne, and while its gallant defender lay suffering in Frederic hospital from amputation of both arms, it was reverently wrapped by Col. Clark and returned to the State House in Boston, with the request that others might be sent them; The 21st had never lost their colors but they had worn them out,

The flag and its brave bearers are alike past their usefulness, save as examples for emulation, and titles of glory for some bright page of our nation's history; and while the one is carefully treasured in the sacred archives of the State, need I more than ask of this noble body to put forth its protecting arm to shelter, cherish and sustain, the other. If guaranty were needed for the private character of so true a soldier, it would have been found in the touching address of his eloquent Colonel (Clark) delivered on Christmas beside the stretcher waiting at the train at Yarmouth to convey its burdened helpless soldier to the care, whither he had been escorted not only by his regiment but his General. The tears which rolled over the veteran cheeks around him were ample testimony of the love and respect he had won from them and long his heart's deepest affections turned around his regiment as the defenders of this nation's security,

A moments reflection will obviate the necessity of any suggestion in reference
May 22nd, 1914.

Hon. Theodore Roosevelt,
Oyster Bay, N.Y.

Dear Sir:

In Dec. 1903, I had passed through the Senate and House, at Salem, Oregon, a Resolution committing the Legislature of the State of Oregon, to your renomination at that time. I am still making another prediction, which you will observe from the inclosed interview. I am still with you, as I have always been, because I have felt that while you have seemed to think that I was guilty of criminal misconduct, someday I am going to convince you that you have been decieved, and I have never had any hard feeling towards you because I have felt that you had to believe what certain parties in Oregon felt in their duty, at all times to impress upon you.

I was a candidate for governor this year, and polled a splendid vote although there were eight other candidates in the field.

I am not writing this letter with any hopes of reward from you, or anything of that kind; I simply believe that you are the man to meet the demand of the country in the next Presidential contest, and I am for you. You will carry Oregon with a big majority. Everything indicates that the sentiment here now, is very strong for you.

Very sincerely yours,

Geo. C. Brownell
\title{Examples on the Integral Calculus}

\section{Chapter I}

\subsection{Elementary Integrals to be Committed to Memory}

(1.)
\[
\int \frac{dx}{x} = \log x.
\]

(2.)
\[
\int \frac{dx}{a + b x^2} = \frac{1}{\sqrt{ab}} \tan^{-1}\!\left(x\sqrt{\frac{b}{a}}\right),
\]

or

\[
\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}\!\left(\frac{x}{a}\right).
\]

(3.)
\[
\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\!\left(\frac{x}{a}\right).
\]

(4.)
\[
\int \frac{-dx}{\sqrt{a^2 - x^2}} = \cos^{-1}\!\left(\frac{x}{a}\right).
\]

(5.)
\[
\int \frac{dx}{\sqrt{x^2 \pm a^2}} = \log \left( \frac{x + \sqrt{x^2 \pm a^2}}{a} \right).
\]

(6.)
\[
\int \frac{dx}{\sqrt{2ax - x^2}} = \mathrm{vers}^{-1}\!\left(\frac{x}{a}\right).
\]

(7.)
\[
\int \frac{dx}{\sqrt{x^2 \pm 2ax}} = \log \left(x \pm a + \sqrt{x^2 \pm 2ax}\right).
\]

(8.)
\[
\int \frac{dx}{x\sqrt{x^2 - a^2}} = \frac{1}{a} \sec^{-1}\!\left(\frac{x}{a}\right).
\]
(8.) To find the volume of a parabolic spindle.

Put $CD=h$, $AB=2c$, and $x=AG$, and $y=FG$.

From the property of the parabola

$$AD^2 : AG\cdot GB :: CD : EG,$$

$$c^2 : x(2c-x) :: h : y,$$

$$\therefore y=\frac{h(2cx-x^2)}{c^2}, \quad \therefore y^2=\frac{h^2}{c^4}(4c^2x^2-4cx^3+x^4).$$

$$\therefore V=\pi \int y^2 \, dx$$

$$=\pi\frac{h^2}{c^4}\int(4c^2x^2-4cx^3+x^4)dx=\pi\frac{h^2}{c^4}\left(\frac{4c^2x^3}{3}-cx^4+\frac{x^5}{5}\right),$$

the volume of $AFH$, since $C=0$; when $x=c$, we have

$$\frac{\pi h^2}{c^4}\left(\frac{4c^5}{3}-c^5+\frac{c^5}{5}\right)=\frac{8}{15}\pi h^2 c=$$

volume of the semi-spindle; and, as we have found the volume of the part $AFH$, if we subtract this from the whole semi-spindle, we shall have the frustum $EHCF$, the double of which will be volume of the whole frustum $EHIL$.

In the same manner as in the last example, we may find the surface of the parabolic spindle.

\begin{figure}
\centering
\includegraphics{figure_1.pdf}
\label{figure_1}
\end{figure}

(9.) In a parabola find the area included between the curve, its evolute, and its radius of curvature.

area parabola $ANP=\int y\,dx$

$$=2\sqrt{a}\int \sqrt{x}\,dx=\frac{4}{3}\sqrt{a}\,x^{3/2},$$

area evolute $AN'P'=\int \beta\,da=\frac{2}{3\sqrt{3}\,a}\int (a-2a)^{3/2} da$

\begin{figure}
\centering
\includegraphics{figure_2.pdf}
\label{figure_2}
\end{figure}
∴ semi-area
\[
= \frac{c(c^2-a^2)}{4a} \sin^{-1} \frac{\sqrt{c^2-a^2}}{\sqrt{c^2-a^2}}
= \frac{c(c^2-a^2)}{4a} \cdot \frac{\pi}{2}
\]

area circle
\[
= \frac{a}{2} \cdot 2\pi b = \pi ab,
\]

area between epicycloid and circle
\[
= \frac{c(c^2-a^2)}{4a} \pi - \pi ab
\]
\[
= \frac{(a+2b)^2\,4b(a+b)}{4a} \pi - \pi ab
\]
\[
= \frac{\pi}{a}(a^2b + 3ab^2 + 2b^3 - a^2b)
\]
\[
= \pi b^2 \left(3 + \frac{2b}{a}\right).
\]

(18.) Find the length of the curve where
\[
x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}},
\]

\[
s = \int \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
= \int \sqrt{1 + \frac{y^{\frac{2}{3}}}{x^{\frac{2}{3}}}} \, dx
\]
\[
= \int \sqrt{\frac{x^{\frac{2}{3}} + y^{\frac{2}{3}}}{x^{\frac{2}{3}}}} \, dx
= \int \frac{a^{\frac{1}{3}}}{x^{\frac{1}{3}}} \, dx
= \frac{3}{2} a^{\frac{1}{3}} x^{\frac{2}{3}}.
\]

Taking it between the limits $x=0$, $x=a$,
\[
s = \frac{3}{2}a.
\]

The whole length of the curve $4 \times \frac{3}{2} a = 6a$.
And
\[
\int \frac{dx}{\sqrt{2ax + x^2}} = \log \{x + a + \sqrt{2ax + x^2}\},
\]

\[\therefore \int \frac{x\,dx}{\sqrt{2ax + x^2}}
= \sqrt{2ax + x^2} - a \log \{x + a + \sqrt{2ax + x^2}\}.
\]

\[\therefore \int \frac{x^2\,dx}{\sqrt{2ax + x^2}}
= \frac{x\sqrt{2ax + x^2}}{2} - \frac{3a}{2}\sqrt{2ax + x^2}
+ \frac{3a^2}{2} \log \{x + a + \sqrt{2ax + x^2}\}.
\]

(3.)
\[
\int_a^0 \frac{x^2\,dx}{(2ax - x^2)^{3/2}}
\]

let $p = x$, \quad $dp = dx$, \quad $dq = \dfrac{x\,dx}{(2ax - x^2)^{3/2}}$,

\[\therefore q = \int \frac{x\,dx}{(2ax - x^2)^{3/2}}
= \int (2a - x)^{-3/2} x^{1/2} \, dx
= \int (2a x^{-1} - 1)^{-3/2} x^{-2} \, dx
\]

\[
= -\frac{1}{2a} \int (2a x^{-1} - 1)^{-3/2} \times (-2a x^{-2}) \, dx
\]

\[
= \frac{(2a x^{-1} - 1)^{-1/2}}{a} = \frac{1}{a} \sqrt{\frac{x}{2a - x}}.
\]
Let
\[
\frac{x^2}{(x^2+1)(x^2+4)} = \frac{A}{x^2+1} + \frac{B}{x^2+4}
\]
\[
x^2 = A(x^2+4) + B(x^2+1).
\]

Let $x = \sqrt{-1}$, or $x^2 = -1$,
\[
\therefore -1 = 3A, \qquad \therefore A = -\frac{1}{3},
\]

\[
\therefore x^2 + \frac{1}{3}(x^2+4) = B(x^2+1),
\]

\[
\therefore \frac{4(x^2+1)}{3} = B(x^2+1), \qquad \therefore B = \frac{4}{3}.
\]

\[
\int \frac{x^2 \, dx}{(x^2+1)(x^2+4)} = -\frac{1}{3} \int \left\{ \frac{dx}{x^2+1} - \frac{4\,dx}{x^2+4} \right\}
\]

\[
= \frac{1}{3} \int \left\{ \frac{dx}{1+x^2} - \frac{dx}{1+\frac{x^2}{4}} \right\}
\]

\[
= \frac{1}{3} \left\{ 2 \tan^{-1} \frac{x}{2} - \tan^{-1} x \right\}.
\]

Or thus,

\[
\int \frac{x^2 dx}{(x^2+1)(x^2+4)} = \frac{1}{3} \int \frac{3x^2 dx}{(x^2+1)(x^2+4)}
\]

\[
= \frac{1}{3} \int \frac{\{4x^2 + 4 - (x^2+4)\} \, dx}{(x^2+1)(x^2+4)}
\]

\[
= \frac{1}{3} \int \left( \frac{4dx}{x^2+4} - \frac{dx}{x^2+1} \right)
\]

\[
= \frac{1}{3} \left( 2 \tan^{-1} \frac{x}{2} - \tan^{-1} x \right).
\]

(5.)
\[
du = \frac{(3x^2 + x - 2)\,dx}{(x-1)^3 (x^2+1)}.
\]
and the integral required is therefore

\[
\log \sqrt{x^2 + 1} + \frac{1}{x^2 + 1} - \frac{1}{4(x^2 + 1)^2},
\]

\[
= \frac{4x^2 + 3}{4(x^2 + 1)^2} + \log \sqrt{x^2 + 1}.
\]

(22.)
\[
\int \frac{x^2 \, dx}{x^4 + 1},
\]

\[
(x^m + 1) = (x^2 - 2x \cos \tfrac{\pi}{m} + 1)(x^2 - 2x \cos \tfrac{3\pi}{m} + 1) \ldots
\]

continued to the factor

\[
(x^2 - 2x \cos \tfrac{m-1}{m}\pi + 1)
\]

when $m$ is an even number.

This gives

\[
(x^4 + 1) = (x^2 - 2x \cos \tfrac{\pi}{4} + 1)(x^2 - 2x \cos \tfrac{3\pi}{4} + 1),
\]

or

\[
(x^4 + 1) = (x^2 - x\sqrt{2} + 1)(x^2 + x\sqrt{2} + 1).
\]

Since, $\tfrac{\pi}{4} = 45^\circ$, $\cos \tfrac{\pi}{4} = \sin \tfrac{\pi}{4} = \tfrac{1}{\sqrt{2}}$, $\cos \tfrac{3\pi}{4} = -\sin \tfrac{\pi}{4}$.

Assume therefore

\[
\frac{x^2}{x^4 + 1} = \frac{Ax + B}{x^2 - x\sqrt{2} + 1} + \frac{Cx + D}{x^2 + x\sqrt{2} + 1};
\]

\[
\therefore x^2 = (Ax + B)(x^2 + x\sqrt{2} + 1) + (Cx + D)(x^2 - x\sqrt{2} + 1).
\]

If $x^2 + x\sqrt{2} + 1 = 0$, \[ x = \frac{\sqrt{-1} - 1}{\sqrt{2}} \] (1).

If $x^2 - x\sqrt{2} + 1 = 0$, \[ x = \frac{\sqrt{-1} + 1}{\sqrt{2}} \] (2).
\[
\therefore \int \frac{dx}{(1+x)\sqrt{1-x-x^2}}
= - \int \frac{dz}{z^2 \cdot \frac{1}{z} \sqrt{1+\frac{1}{z}-\frac{1}{z^2}}}
\]

\[
= - \int \frac{dz}{\sqrt{z^2+z-1}}
= - \int \frac{dz}{\sqrt{\left(z+\tfrac{1}{2}\right)^2-\tfrac{5}{4}}}
\]

\[
= - \log \left\{ \sqrt{z^2+z-1}+z+\tfrac{1}{2} \right\}
\]

\[
= - \log \left\{ \frac{\sqrt{1-x-x^2}}{1+x}+\frac{1}{1+x}+\tfrac{1}{2} \right\}
\]

\[
= - \log \left\{ \frac{2\sqrt{1-x-x^2}+x+3}{2(1+x)} \right\}
\]

\[
= \log \left\{ \frac{2(1+x)}{2\sqrt{1-x-x^2}+x+3} \right\}
\]

\[
= \log \left\{ \frac{2(1+x)(x+3)-2\sqrt{1-x-x^2}}{x^2+6x+9-4+4x+4x^2} \right\}
\]

\[
= \log \frac{2(1+x)\left(x+3-2\sqrt{1-x-x^2}\right)}{5(1+x)^2}
\]

\[
= \log \frac{x+3-2\sqrt{1-x-x^2}}{1+x}+c.
\]

(36.)

\[
\int \frac{dx}{\sqrt{1+2x-x^2}}
= \int \frac{dx}{\sqrt{2-(1-x)^2}}
\]

\[
= - \int \frac{d(1-x)}{\sqrt{2-(1-x)^2}}
= \cos^{-1} \frac{1-x}{\sqrt{2}}.
\]
(39.)
\[
\int \frac{\sqrt{x}\,dx}{\sqrt{a^3 - x^3}}.
\]
Let \(z^2 = x^3\).

\[
x = z^{\frac{2}{3}}, \qquad dx = \frac{2}{3}z^{-\frac{1}{3}}dz, \qquad \sqrt{x} = z^{\frac{1}{3}}.
\]

\[
\therefore \int \frac{\sqrt{x}\,dx}{\sqrt{a^3 - x^3}} = \frac{2}{3} \int \frac{dz}{\sqrt{a^3 - z^2}} = \frac{2}{3} \sin^{-1} \frac{z}{a^{\frac{3}{2}}}
\]

\[
= \frac{2}{3} \sin^{-1} \frac{x^{\frac{3}{2}}}{a^{\frac{3}{2}}}.
\]

Let \(\theta = \sin^{-1} \dfrac{x^{\frac{3}{2}}}{a^{\frac{3}{2}}}\), \quad \(\sin \theta = \dfrac{x^{\frac{3}{2}}}{a^{\frac{3}{2}}}\).

\[
\tan \theta = \frac{\sin \theta}{\sqrt{1-\sin^2 \theta}} = \frac{x^{\frac{3}{2}}}{\sqrt{a^3 - x^3}}.
\]

\[
\therefore \int \frac{\sqrt{x}\,dx}{\sqrt{a^3 - x^3}} = \frac{2}{3} \tan^{-1} \sqrt{\frac{x^3}{a^3 - x^3}}.
\]

(40.)
\[
\int \frac{dx}{(2ax + x^2)^{\frac{3}{2}}} = \int \frac{dx}{\big((x+a)^2 - a^2\big)^{\frac{3}{2}}}.
\]

Let \(x + a = \frac{1}{z}\), \quad \(dx = -\frac{dz}{z^2}\).

\[
= - \int \frac{dz}{z^2\left(\frac{1}{z^2} - a^2\right)^{\frac{3}{2}}} = - \int \frac{z\,dz}{(1 - a^2 z^2)^{\frac{3}{2}}}.
\]
∴
\[
\int \frac{dx}{(a^2+x^2)^n}
= \frac{1}{a^2}\frac{x}{(2n-2)(a^2+x^2)^{n-1}}
- \frac{1}{a^2(2n-2)}\int \frac{dx}{(a^2+x^2)^{n-1}}
+ \frac{1}{a^2}\int \frac{dx}{(a^2+x^2)^{n-1}}.
\]

\[
= \frac{1}{a^2(2n-2)}\frac{x}{(a^2+x^2)^{n-1}}
+ \frac{2n-3}{a^2(2n-2)}\int \frac{dx}{(a^2+x^2)^{n-1}}.
\]

Also for
\[
\int \frac{x^m\,dx}{(a^2+x^2)^n}
= \int x^{m-1}dx\,\frac{x}{(a^2+x^2)^n}.
\]

We have, if $p=x^{m-1},\; dq=x(a^2+x^2)^{-n}dx$:

\[
\int \frac{x^m\,dx}{(a^2+x^2)^n}
= \frac{1}{2-2n}\frac{x^{m-1}}{(a^2+x^2)^{n-1}}
+ \frac{m-1}{2n-2}\int \frac{x^{m-2}\,dx}{(a^2+x^2)^{n-1}}.
\]

\[
du=(a^2-x^2)^{\frac{n}{2}}dx
\]

\[
(a^2-x^2)^{\frac{n}{2}}=a^2(a^2-x^2)^{\frac{n-2}{2}}-x^2(a^2-x^2)^{\frac{n-2}{2}};
\]

\[
\therefore\; u=a^2\int (a^2-x^2)^{\frac{n-2}{2}}dx-\int x\cdot x\,(a^2-x^2)^{\frac{n-2}{2}}dx
\]

\[
= a^2\int (a^2-x^2)^{\frac{n-2}{2}}dx+\frac{x(a^2-x^2)^{\frac{n}{2}}}{n}-\frac{u}{n};
\]

\[
\therefore\; u=\frac{x(a^2-x^2)^{\frac{n}{2}}}{n+1}+\frac{na^2}{n+1}\int (a^2-x^2)^{\frac{n-2}{2}}dx.
\]
Integration gives,

\[
\frac{dx}{dt} = C_1, \qquad \frac{dy}{dt} = -gt + C_2.
\]

When $t=0$, $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ are the components of $v_0$. Hence

\[
C_1 = v_0 \cos \alpha, \qquad C_2 = v_0 \sin \alpha,
\]

and

\[
\frac{dx}{dt} = v_0 \cos \alpha,
\]

\[
\frac{dy}{dt} = v_0 \sin \alpha - gt.
\]

Integrating again, we get

\[
x = v_0 t \cos \alpha,
\]

\[
y = v_0 t \sin \alpha - \tfrac{1}{2}gt^2,
\]

the constants being zero because $x$ and $y$ are zero when $t=0$.

\begin{figure}
\centering
\includegraphics{figure_4.pdf}
\caption{Fig. 4.}
\label{figure_4}
\end{figure}

\section{Curves with a Given Slope}

If the slope of a curve is a given function of $x$,

\[
\frac{dy}{dx} = f(x),
\]

then

\[
dy = f(x)\,dx
\]

and

\[
y = \int f(x)\,dx + C
\]

is the equation of the curve. Since the constant can have any value, there are an infinite number of curves having the given slope. If the curve is required to pass through a given point $P$, the value of $C$ can be found by substituting the coordinates of $P$ in the equation after integration.

\begin{figure}
\centering
\includegraphics{figure_5.pdf}
\caption{Fig. 5.}
\label{figure_5}
\end{figure}