. A new treatise on the elements of the differential and integral calculus . *^ Vl H a a^ == sin. at ^= sin.^ – • a a. X But, since – a . a Tt 1 . , .r 1 a rr Bin —[-cos.^- = -, sin. –^ cos.^ -: X X

Calculus – Definite Integrals
Calculus – Definite Integrals

. A new treatise on the elements of the differential and integral calculus . *^ Vl H a a^ == sin. at ^= sin.^ – • a a. X But, since – a . a Tt 1 . , .r 1 a rr Bin —[-cos.^- = -, sin. –^ cos.^ -: X X 2 a ii a x a – hence, throwing into the constant ot mtogralion, wo may write /dx 1 , (7 —7 o ., = – cosr^ —xx^ — a^ a x 332 INTEGRAL CALCULUS. 17. f , =^ Makea;=-: ,, dx ^=. x, and r ^^ if dt 17/,, ITT^ hf^ , |1^1 ly^ + V^^ ct x yix^ al a X ax 11 X 1 X a .a a + Va^iaj^ a a + Va^ioj^by including – ?a in the constant of integration. Cli •^ ic^ — a^ 2a •^ a? — a x — a/1 r dx 1 r d

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. A new treatise on the elements of the differential and integral calculus . *^ Vl H a _ a^ == sin.~ at ^= sin.~^ – • a a. X But, since – a . a Tt 1 . , .r 1 a rr Bin —[-cos.~^- = -, sin.~ –^ cos.~^ -: X X 2 a ii a x a – hence, throwing into the constant ot mtogralion, wo may write /dx 1 , (7 —7 o ., = – cosr^ —xx^ — a^ a x 332 INTEGRAL CALCULUS. 17. f , =^ Makea;=-: , , dx ^=. x, and r ^^ if dt 17/, , ITT^ hf^ , |1^1_ ly^ + V^^ ct x yix^ al a X ax 11 X 1 X a .a a + Va^iaj^ a a + Va^ioj^by including – ?a in the constant of integration. Cli •^ ic^ — a^ 2a •^ a? — a x — a/1 r dx 1 r dx 2a*^ X — a 2a^x-]-a 1 7/ ^ 1 7/ N 1 7 ^ — ^ 2a ^ 2a ^ ^ ^ 2a a; + a If X is less than a, then r dx r dx 1 r/ dx dx ^ x^ — a^ *^ a^ — x^ 2a^ a — x a 1 7 N 1 7 N 1 J a — X ^ — l(a — x) — -—l(a — x) z= —- I —;—:2a ^ ^ 2a ^ ^ ^ 2a a + a? and. 19. pSuppose q — to be negative, and make MISCELLANEOUS EXAMPLES. 333 then, by the last example, /dx r dt ^ nf, /2 I ^n, . I ^y »/ /■- r/, ^ dx r dt 1 7 ^ — ^ cc^ -[-px — q ^ t — a^ 2a t — a 1 2x-{-p — /4tq — p V4:q—p^ 2x–p-{-/4:q — p p P If g —7 ^^ positive, then q — — =: a^; and dx _ r ^^ __ ^ + -1 ^ /dx /* i^;^ 4- z>a3 4- r7 J X- –px — q J t^ -^ a^ = – tan. tan. .1 2x +p^/q—p^ ^/^q — p^ I , mp , 7np 20. f-J!pl^dx = , i ^ ^ d. J x^ –px — q I x^ –px — q 2 •^ fl:;2 _j_ p^ _|_ g ^ 2 /^ x^ –px-{- q The integral of the first term is -^ I{x^ -{-px -f- q), and that of the second is found by Ex. 19. r dx rsiu.x , r dcos.x 21. J -. =J -.-r-dx = —J -. ^ sm. X ^ sin.-U/ *^ 1 — COS.-a; Make cos. x =.t; then by Ex. 18 : but 1, 1-4- COS. X 7 /I — cos. xk , , .r ^ = M , .— I == I tan. – • 2 1 — cos. X 1 + COS. X. dx r COS. xdx /* (7 sin. .t* r (tx^ _ /* COS. xdx _ /* (/ sui. .t*J COS. X J COS, a; J 1 — sin.^ a; 334 INTEGRAL CALCUL

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