(sin x)3 (cos x)* dx = [{1— (cos x)2} (cos x)1 sin x da

Ex. 2.

S(sin

72.] When neither of the three above-mentioned conditions as to m and n is fulfilled, we must have recourse to integration by parts, and proceed as follows:

(sin x)TM cos x dx (cos x)"−1 ;

f(sin x)" (cos x)" dx = = f(sin x) on comparing which with the typical form ·Ju dv,

let dv (sin x)" cos x dx, u = (cos x)"-1;

which is an useful form when m is negative and n is positive.

which is useful when n is negative and m is positive.

Also the last term of the right-hand member of (46) may be written in the form,

(cos x)" (sin x)−2 (cos x)2 dx

= f(cos x)" (sin x)m−2 {1 — (sin x)2} dz

= f(cos x)" (sin x)"-2 dr-f(cos x)" (sin a')" dæ ;

substituting which in (46) and reducing, we have

Similarly may other formulæ be constructed; but the form of the element to be integrated will usually suggest various modifications by which it may be transformed into some known integral.

-5

(tan x)5

-3

= [ { (tan x)−5 + 2 (tan x)−3 + (tan x)−1} d. tan æ

73.] Integration of (tan x)" dx, and of (cot x)" dx.

These formulæ give definite results for even values of n, but ultimately fail when n is odd; in which cases however, by (18) ́and (19), Art. 67,

Star

tan x dx = log sec x,

= sinx- -n

cos x + (n−1) √xn−2

=x" sin x+nx”-1 cos x — n (n−1)

cos x − n (n − 1) fxn−3.

Similarly it may be shewn that

Ex. 1.

cos x dx

cos x dx. (50)

sin x dx =

sin x dx. (51)

x3 cos x dx = x3 sin x+3x2 cosx-6x sinx-6cosx.

And hence we may integrate infinitesimal elements of the forms

m

m

x" (sin x) dx, x" (cos x) dx;

(53)

for if (sin x) and (cos x)" are expressed in terms of the sines and cosines of the multiple arcs, by means of Arts. 62, 63, Vol. I, then each term of the integral will be of one of the forms (52), and may be integrated accordingly.

75.] Integration of eax (cos x)" dx, and of ea* (sinx)" dx.

- 1 f {(cos x)" — (n − 1) (cos x)"−3 (sin x)2} eax dx

76.] Integration of ear cos nx dx, and of ear sin nx dx.

4+ a2 a

These results may also be obtained as follows, by expressing sin nr and cos na in terms of their exponential values :

=

=

2

eaz enx dx

x

eax

2 (a2 +n2) { a (e"√=12 + e ̄*√=1)−n√/−1 (e"√=1z_e ̃»√=1)}

-n

{a cos nx+n sin nx}.

... §,+S2√ −1 = fea2 {cos næ + √/−1 sin næ} dæ

77.] Integration of f(x) sin-1x dx, f(x) tan-1x dx, &c. Integrals of these forms must be determined by integration by parts; the method is best exhibited by examples such as follow: