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INTEGRAL CALCULUS.

CHAPTER I.

INTEGRATION OF FUNCTIONS OF ONE VARIABLE.

The fundamental formulæ to which all integrals are reduced are the following.

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(a) sdx an

n + 1

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1

log 2 a

(a),

Xta

-1

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1

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dx

dx (e)

sin and (a’ – 2*)!

(a* – ) dx

|(x* + a')} + x

= log (x* = a*)

dx (8)

x x– a)

dx ) (?

(a + 22

)} + a ar (i) Sdx a

or sd & 4* log a

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sec

a

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By simple algebraic transformations we may frequently put an integral into a shape in which one or other of the preceding formulæ is at once applicable. dx xn-1 dx nbx"

log (a + bx;").
nb

a + bxn nb
dx
dx

da-r)
{a’ -(a–x)"}}

1

(1) Saxo

=

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a + bor

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-S

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which is integrated by (c) or by (d) according as 4ac 6? > 0 or <0. Hence we have

dx
2

X + 1
3!

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The integral Sta+ bx + cauty

dw

is reduced to

dx

ਦੇ

2

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(-)"

dx

1

or to
4ac-6?)

c
(4ac+b*

6
+
402

4c according as the upper or lower sign of c is taken ; and these are of the forms (f) or (e) respectively. Hence


s* log {2x + 1 + 2 (1 + x + x?)}}.

dr
(12)
(1 + 2x – w*)

23
dx
(13) s

= log {2x – 1 + 2 (v* – X – 1)"}.
(x2 – X – 1)

dw
(14) S
(1
202)2

5)
dx ax + b)
The integrals

may be split into
m2 + px + 9
ap
dc

(2x + p) dx

si X* + px + 9

2

° + px + 9 the first of which is integrable by (c) and the second by (6). Hence

x da

log (a + 25x + 2*)!

2 x + 1

= sin -1

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a +

(15) Sæt +260 + m2

X + b

tan

}

tan-1

W + 1 2)

6

(a* b)] lla? – bo)!) (2x – 1) da

3 (16) S

· log (x2 + 2x + 3)
x2 + 2x + 3

22
(1
X cos ) da

cos
)

sin A tan-1 cos O + x

sin e

- cos 0 log (1 – 2w cos 0 + a'?). In this example the numerator may be readily split by observing that 1 = cos" 0 + sive.

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By multiplying the numerator and denominator of a fraction by the same quantity it may frequently be split into integrable parts or reduced to an integrable shape.

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dx x-2
d ()

(29) Sola+60+ copy) = S(aw-+62-" + c)?

--S (ax-+62-2 + c)? which is of the same form as Sca

(30) Sant 1-log{+++5+6)}

da
(a + bx + cxo)?

Therefore

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sin-' (2)

dx

(31) Se cont (32) Sta+ bx + cxo)? (33) Sta+bø + cx*)

2 (20x + b)
(4ac - b) (a + bx + cx®)! *

2 (2a + b)
(4ac - b*)(a + bx + cw*)

x dx

* X

(34) The integral sdx

(1 + x)

can be split into

2

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}

1

1 *

(1 + x) and as the second term within the brackets is the differential

d 1 of the first, it is equivalent to Sdx €". ; and therefore

d x 1 + x

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de-1

(37) Silver - Site

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