In these integrals the radius of the circle is unity ; but
as it is frequently desirable to integrate differentials in
which the radius is r instead of unity, we shall exhibit a
few of those which occur most frequently to that radius.
Thus,






Ex.53. / . - =sin.-' M+C.
Ex.54. / — . ,- ==cos.-' M+C.
r Rdu . ,

/ , „ ==Yersin.~' m+C.
Ex.55. , ,__

V2Rm— M^



'• J Rz+m2^



Ex. 56. J j^2^M2 =tan.-' i«+C.
Ex. 57. Integrate (?M=sin. a; cos. a; . dx.
Put sin. »;=:«/

then, by differentiating, cos. xdx=dv.
Substituting for sin. x and cos. xdx their yalues, we
have

L2



250 INTBGEAL CALCULUS.
1

or '^=^''-'

1.
=- sin.°a!+C.
Ex. 58. If du=sm.? X COS. xdx,

then ■ M=i sin.^ x+C.
Ex. 59. If du —sm? X COS. xdx,

3

then ^< = f sin.^ as + C.
Ex. 60. Integrate. cZm= (tan." e+tan.°+'' d)dQ,
=tan.° e(l+tan.2 B)dB.
Put tan. ft=wy

then, by differentiating, we have
(l+tan.2e)cZ9=(?w,
and substituting, du—v'dv;

:. u=—r-7 tan.°+' 0+C.
n-\- 1

Ex. 61. If (?M=(tan.5e+tan.'' e)c?0,

then M=|- tan.^ e+C.
Ex. 62. Integrate du=sva. 2xdx.
By (Trig., Prob. XVI.), sin. 2a;=2 sin. x cos. x;

' .: du—2 sin. x cos. cc^a;.
M=sih;?£e+C.
Ex.63. If du={cot.^e+_cot.^ d)de,

u=—icot.^8+G.
Ex. 64. Integrate ,

Put Vx'^dzo^—v;

then aj^ia^—^a.
hence, by differentiating, xdx=vdv.
Putting this into a proportion,

x:v:: dv : dx,
or, by composition, x+v:v:: dx-\-dv : dx;
dx+dv dx dx
■• x+v ~ V ~"-v/a;2±a2'

/dx , , \
■ =:log. (x4-v),

or, substituting for v its value,



:log. («+ VV±a^).
INTEGRATION OF CIECULAE FUNCTIONS. 251

Ex. 65. Integrate , :.
Va;2±2aa3

Put V'a;2±2aa;=uy

then x'^±2ax=v'^,

xdza—Vv^+c^ ;

vdv

.: dx— I — ^=.
