INTEGRAL CALCULUS.

CHAPTER I.

INTEGRATION OF FUNCTIONS OF ONE VARIABLE.

THE fundamental formula to which all integrals are reduced are the following.

=

- 1

a

1

a

and

± a2)3

dx

(a2 - x2)

sec-1

log

a

a

a2 ± x2) 1⁄2 + a

(i) fda a' -, or fdxe"==,
a* =
log a

cos ma, and fdx cosmx = −

(1) fdx (sec x)2 = tan x.

m

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.

dxx

=

=

2 a2

log (a + bx”).

-S

=

-1

d (a−x) {a2 - (a-x)2}}

+ #+ q).
a}.

= sin-1

tan

༢/

d (x2 - a3) $ b2 − (x2

(©) √ {(x2 = a') (b2 - a') }} = √ {8 - a2 - (a' - a') } 1

4ac-b2'

4c2

which is integrated by (c) or by (d) according as 4ac - b2 >0 or <0. Hence we have

(8) - log (2-1+51)

1

=

dx

dx

(10) + 3 + 2.log(2+1))

(x

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

dx

Hence

(11) J (1 + x + x2) = log (2x+1+2 (1 + x + x2)}.

dx

23

(x2 − ∞ − 1)4 = log {2x − 1 + 2 (x2 − x − 1)} } .

dx

(12) S

dx
(1 + 2x − x2)§

= sin-1

(13) s

(14) S

2+ p + q dx (ax+b)

53

may be split into

x2 + px + q

the first of which is integrable by (c) and the second by (b).

In this example the numerator may be readily split by observing that 1 = cos2 0 + sin2 0.

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.

dx

dx x-2

(29) √ x (a + bx + ca2)1 = √ (ax22 + bx ̄1 + c)}

=

-2

d (x-1)

dx

which is of the same form as (a + bx + ca2)

Therefore

=

log {2

(34) The integral sda

=

=

00

2 + x + 2 (1 + x + 1 ) })})

sin-1 (1)

2 (2cx + b)

and as the second term within the brackets is the differential