DERIVATIVES AND INTEGRALS 
3. Show that the integral of 1 /(a + b cos #), where a + b is positive, may 
be expressed in one or other of the forms 
2 , f /fa-h\\ 1 , (V(6 + a) + W(ft-«0] 
\J{a 2 -62) arctan Y V [a + b)] ’ J{b 2 -a 2 ) l0g W(b + a)-tJQ) — d)] ’ 
where ¿ = tan|#, according as a 2 > i 2 . If a 2 =b 2 the integral reduces to a 
constant multiple of that of sec 2 1# or cosec 2 I #, and its value may at once 
be written down. Deduce the forms of the integral when a + b is negative. 
4. Show that if y is defined in terms of # by means of the equation 
(a + bcosx){a- bcosy)=a 2 -b 2 , 
where a 2 > b 2 , then as x varies from 0 to n one value of y varies from 0 to tt. 
Show also that 
J{a 2 - b 2 ) sin y sin x dx _ sin y 
sin#— a _£ COS y ’ a+bcosxdy a — bcosy' 
and deduce that, if 0 < x < tt, 
f dx 1 "(a cos x+ b\ 
J a + b cos x (a 2 - b 2 ) <U C C0S \« + b cos x) ' 
Show that this result agrees with that of Ex. 3. 
5. Show how to integrate 1 ¡{a + b cos x + c sin x). [Express b cos x + c sin x 
in the form J{b 2 +c 2 ) cos (#-a).] 
6. Integrate (a + bcosx+csinx)/{a+l3cosx + y sin#). 
[Determine X, p., v so that 
a + b cos x + c sin x=\ + y,(a+P cos #+y sin #) + v ( — /3 sin # + y cos #). 
Then the integral is 
ix# + vlog(a + /3cos# + ysin#) + X —2 • .] 
r o\ r / j a + /3cos#+ysm# J 
7. Integrate l/(5+3cos#), 1/(3-5cos#), 1/(2-sin#), 1/(1-cos# + 2sin#), 
(5 + 3 cos # - 7 sin #)/(! 1 - cos # + sin #), 
8. Integrate 1 /(a cos 2 # + 2b cos # sin x + c sin 2 #). [The subject of inte 
gration may be expressed in the form 1/(^4 + B cos 2# + C sin 2#), where 
A=\{a+c), B = \(a- c), C—b: but the integral may be calculated more 
simply by putting tan x = t, when we obtain 
f sec 2 # dx _ i dt 
Ja + 2Z)tan#+ctan 2 # J a + 2bt + ct 2 
126. Integrals involving arc sin x, arc tan x, and log x. The 
integrals of the inverse sine and tangent and of the logarithm can 
easily be calculated by integration by parts. Thus 
r f oo doo 
I arc sin xdx = x arc sin x — —m = on arc sin x+ — x 2 ),