36 
CALCULUS 
Hence, finally, 
i'xtan-dx = tan-?î±ü?_ log [o , + (a , + 
J cl* 2 a 2 4 
CL* 2 a 
The student may like to evaluate 
Çx? tan- 1 «!^ 2 
J a 2 
dx. 
EXERCISES ON CHAPTER 
Evaluate the following integrals, using the methods, but not the 
formulas, developed in the text. 
r xdx 2 Ç(e T - e~*) 2 dx 3 
J Va 2 — a? 2 .y e 1 
J 
/ 
/• 
cos æ 2 da;. 
4 i sin(7rV.9 — 8) ds 5 / 2 cos ax — sin ^ g /*i 
\/s CL -j- /3 fj 
e * 
dx. 
e~ x dx 
e z + e~ x 
/Vïtss* 1 9 - 
f* dx 
J nr. 1 iter nr% 
X log X 2 
cos ddd 
cos 2 9 
13. 
15 
18. 
20 
22 
25 
J -—• 11. / iccot 2 a;dic. 12. / ( 
5 — sm 2 0 J J 
/ ædæ /*, 
. • 14. / log (1 —(- a? —f— a? 2 ) da?. 
(a 2 + a: 2 ) V4 -f- x 2 J 
f T- • 16. j x 1 tan -1 x dx. 17. f J?-—.. . 
J 1 + tan 9 J J (i _ x y 
f dx 19 r d9 
J (1 + x 2 ) tair 1 x ' J sin 2 9 - 2 cos 2 0* 
a; sin -1 ^^ dx. 21. J e~ x * (2 x — 3 x?) dx. 
j*cos 5 ddd. 23. y*x tan ttx 2 dx. 24. j x ^ x , 
fsing-cose dd 26 r 
J 1 + smd +• cos 0 J 
dd. 26. I sinpx cos qx dx. 
28 r (1 + a cos 0) dO /* 
J (1 + a 2 )cos 2 0 +1 29, J ( a + x ) lo S (* + Va 2 - x 2 ) dx. 
30. Çsin mO sin nO dO. 31. /* V« 2 + l» 1 + a 2 sin 2 9 ^ 
J , J cos 9 
cos 9