64.] Examples of integration of various logarithmic functions.

SECTION 2.-Integration of Circular Functions.

65.] Integration of the fundamental circular functions.

66.] And all the preceding formulæ are of course true when for x any function of x, say f(x), is substituted, provided that da is replaced by f'(x) dx.

Jsin (mx+

Thus

1

=

1

m

1

[sin(x3) x2 dx = {} [sin(x3) d. a3 = − } cos (x3).

3

cos (mx2+nx+p) (2mx+n) dx = sin(mx2+nx+p).

dx

= cot x.

(15)

1

sec mx tan mx dx =

sec mx;

(16)

m

1

m

67.] Integration of other circular functions by transformation into the fundamental forms.

dx

2

2

2

a { (cos ~)2 + (sin ~)" } + ¿ { (cos ~)3 – (sin ~)"}

2

a+b+ (a—b) (tan 2)2

(a) let a be greater than b; then the last expression becomes

(26)

+ (tan)

Similarly may the integrals of the product of three or more sines

and cosines be determined.