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IntegralSummary - Indefinite Integral Table Note u v and w...

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Unformatted text preview: Indefinite Integral Table Note: u, v, and w are functions of x. a, c, n, are constants. All trigonometric functions use radians. A constant must be added 4. to the result of every integration. General and Basic Integrals 1. ∫ a f ( x ) dx = a ∫ f ( x ) dx 2. ∫ ( u ± v ) dx = ∫ u dx ± ∫ v dx 3. ∫ u dv = uv – ∫ v du g′(x ) 5. ∫ ------------ dx = ln g ( x ) g(x) { [ g ( x) ]r + 1 } ⁄ ( r + 1 ) r ≠ 1 6. ∫ [ g ( x ) ] r g ′ ( x ) dx = ln g ( x ) r=1 x n + 17. ∫ x n dx = ----------n+1 dx 8. ∫ ---- = ∫ x –1 dx = x dx 9. ∫ ---- = ∫ x –n dx = xn SIN 1 = – -- cos ax a 1 1 1 1 sin 2 ax 2 -- x – -- sin 2 x 4. ∫ sin 2ax dx = -- x – -- ---------------3. ∫ sin x dx = 2 4 2 4a 3 3 13 4 5. ∫ sin x dx = – -- sin x cos x – -- sin x cos x + -- x 8 8 4 n–1 1 n–1 n n–2 6. ∫ sin x dx = – -- sin x cos x + ----------- ∫ sin x dx n n 1. ( n ≠ –1 ) ln x (x ≠ 0) 5. n www.et.byu.edu/~jww8 n–2 1 n–2 n–2 --------------------∫ csc x dx = – n – 1 csc x cot x + n – 1 ∫ csc x dx 1. © 2002 by Jon Wittwer 1 2 2c. ∫ ----------------------------- dx = – -----------------ax 2 + bx + c 2 ax + b Integrals Involving a 2 ± b 2 x 2 –1 1 1 bx 1. ∫ --------------------2 dx = ----- tan ----a2 + b2x ab a b 2 – 4 ac = 0 ∫ 2 a + bx dx = ----- ( a + bx ) 3 / 2 3b ∫ csc –1 x dx = x csc x + ln x + x – 1 –1 2 Combined Trig Functions 2 1. ∫ sin x cos x dx = ( sin x ) ⁄ 2 cos ( a – b ) x cos ( a + b ) x 2. ∫ sin ax cos bx dx = – ---------------------------- – ---------------------------2(a – b) 2(a + b) 3. 2 2. ∫ x a + bx dx = ----------2 ( 3 bx – 2 a ) ( a + bx ) 3 / 2 15 b 2 x n ( a + bx ) 3 / 2 2 an 3. ∫ x n a + bx dx = --------------------------------- – ----------------------- ∫ x n – 1 a + bx dx b(2n + 3) b (2 n + 3 ) x 2 ------------------ dx = -------2 ( bx – 2 a ) a + bx 4. ∫ 3b a + bx xn 2 x n a + bx 2 an - x n – 1 5. ∫ ------------------ dx = --------------------------- – ----------------------- ∫ ------------------ dx b ( 2 n + 1 ) b ( 2 n + 1 ) a + bx a + bx 1 a + bx – a ------ ln -------------------------------- for ( a > 0 ) a a + bx + a 1 6. ∫ ---------------------- dx = –1 x a + bx 2 --------a + bx - tan -------------- for ( a < 0 ) –a –a b(2n – 3) 1 1 a + bx 7. ∫ ------------------------ dx = – ------------------------------ – ----------------------- ∫ ----------------------------- dx a ( n – 1 ) x n – 1 2 a ( n – 1 ) x n – 1 a + bx x n a + bx 1 a + bx 8. ∫ ------------------ dx = 2 a + bx + a ∫ ---------------------- dx x x a + bx b ( 2 n – 5 ) a + bx a + bx ( a + bx ) 9. ∫ ------------------ dx = – ------------------------------ – ---------------------- ∫ ------------------ dx xn a ( n – 1 ) x n – 1 2 a ( n – 1 ) xn – 1 3 /2 –1 1 2 c ( ax + b ) 10. ∫ -------------------------------------- dx = --------- tanh ---------------------a ( cx + d ) ax + b cx + d ac ( a > 0, b > 0 ) –1 1 a + bx 11bx 2. ∫ -------------------- dx = ----- tanh ----- = --------- ln -------------- 2 ab a – bx ab a 2 – b 2 x2 a ∫ sec x tan x dx m n = sec x m+1 4. ∫ csc x cot x dx n–1 n+1 = – csc x for ( a > 0, b > 0 ) 3. sin x cos x n – 1 m n–2 5a. ∫ sin x cos x dx = ------------------------------------- + ------------- ∫ sin x cos x dx m+n m+n ∫ a2 x bx b2x2 a 2 + b 2 x 2 dx = -- a 2 + b 2 x 2 + ----- ln ----- + 1 + --------- 2 2b a a2 ∫ sin m sin x cos x m – 1 n m–2 n x cos x dx = – ------------------------------------- + ------------ ∫ sin x cos x dx m+n m+n m–1 –1 1 1 1 b 6. ∫ ----------------------------------------- dx = ----------------------- ln tan -- cx + tan -- 2 a sin cx + b cos cx a c a2 + b2 x1 – n ----------- ( n ≠ 1 ) 1–n ∫ sin x dx = – cos x 2. ∫ sin ax dx 7. ∫ x sin x dx = sin x – x cos x sin ax x cos ax 8. ∫ x sin ax dx = ------------- – ----------------2 a a 9. ∫ x sin x dx = – x cos x + 2 x sin x + 2 cos x 2 2 Integrals Involving e x e ax 1 F(u) F ( e a x ) dx = -- ∫ ----------- du, u = e a x ∫ au 1 1. ∫ e x dx = e x 2. ∫ e ax dx = -- e ax 3. ∫ xe x dx = xe x – e x a ax a x dx = e - ( ax – 1 ) n e x dx = x n e x – n x n – 1 e x dx ----4. ∫ xe 5. ∫ x ∫ a2 x 1dx 6. ∫ ------------------ = -- – ----- ln ( a + be cx ) a ac a + be cx a + ex ax a – b 7. ∫ ------------- dx = ----- – ----------- ln ( b + e x ) b + ex b b a + be x + ce 2 x ax a – bd + cd 2 8. ∫ -------------------------------- dx = ----- + ce x – ----------------------------- ln ( d + e x ) d + ex d d x 1 ex + 1 -x 9. ∫ ------------------- dx = – -- ------------2e – 1 ( e x – 1 )2 10. 2 1 1 bx b 2 x4. ∫ ------------------------- dx = -- ln ----- + 1 + --------- b a a2 a2 + b2x2 2 –1 a x bx 5. ∫ a 2 – b 2 x 2 dx = -- a 2 – b 2 x 2 + ----- sin ----2 2b a 1 1 –1bx 6. ∫ ------------------------ dx = -- sin ----b a a 2 – b 2 x2 Integrals Involving 1. 2 ax – x 2 ∫ –1 a2 x–a x 2 ax – x 2 dx = ---------- 2 ax – x 2 + ---- cos 1 – -- 2 2 a –1 a3 2 x 2 – ax – 3 a 2 x 2. ∫ x 2 ax – x 2 dx = ---------------------------------- 2 ax – x 2 + ---- cos 1 – -- 6 2 a 2 ax – x 2 3. ∫ ----------------------- dx = x –1 x 2 ax – x 2 + a cos 1 – -- a Integrals Involving a 2 – x 2 a2 – x2 ∫F( ∫ a 2 – x 2 ) dx = a ∫ F ( a cos u ) cos u du, x = a sin u –1 2 ax – x 2 2 2 ax – x 2 x 4. ∫ ----------------------- dx = – --------------------------- – cos 1 – -- x2 x a –1 x 1 5. ∫ ----------------------- dx = cos 1 – -- a 2 ax – x 2 –1 x x 6. ∫ ----------------------- dx = – 2 ax – x 2 + a cos 1 – -- a 2 ax – x 2 –1 3 a2 x2 x + 3a x 7. ∫ ----------------------- dx = – -------------- 2 ax – x 2 + ------- cos 1 – -- 2 2 a 2 ax – x 2 1 1 - ----------1. ∫ --------------- dx = ----- ln x + a a2 – x2 2a x – a 2. a x x a 2 – x 2 dx = -- a 2 – x 2 + ---- sin -- a 2 2 2 –1 10. ∫ x sin x dx = – x cos x + n ∫ x n n n–1 cos x d x 11. 12. ∫ sin –1 x dx = x sin x + 1 – x 2 –1 ∫ 1 2 1 + e ax + 1 1 + e ax dx = -- 1 + e a x + -- ln ---------------------------a a 1 + e ax – 1 1 –1 –1 ∫ sin ax dx = x ( sin ax ) + -- 1 – a 2 x 2 a 1 dx -- cot ax 13. ∫ --------------- = – 2 a sin ax sin ( a + b ) x sin ( a – b ) x 14. ∫ sin ax sin bx dx = – --------------------------- + --------------------------2(a + b) 2(a – b) 1 π ax 115. ∫ ---------------------- dx = -- tan -- – ----- 4 2 a 1 + sin ax COS 1 = -- sin ax a 1 1 1 1 sin 2 ax 2 2 3. ∫ cos x dx = -- x + -- sin 2 x 4. ∫ cos ax dx = -- x + -- ---------------2 4 2 4a 3 3 1 4 3 5. ∫ cos x dx = -- cos x sin x + -- cos x sin x + -- x 8 8 4 n–1 1 n n–1 n–2 6. ∫ cos x dx = -- cos x sin x + ----------- ∫ cos x dx n n 1. x 1 2 11. ∫ ------------------- dx = -- ln ( e x + a 2 – a ) – -a a ex + a2 Integrals Involving a x xn ax xnax n 1. ∫ a x dx = ------2. ∫ x n a x dx = --------- – ------- ∫ x n – 1 a x dx ln a ln a ln a n–3 2 n – 4 ----------------2 - n–2 3 xn -x - dx 3. ∫ ----------------- dx = -------------- x x + 1 – -------------- ∫ 2n – 1 x3 + 1 2n – 1 x3 + 1 n – 3 xn – 4 xn xn – 3 4. ∫ ----------------- dx = ----------- x 4 + 1 – ----------- ∫ ----------------- dx n – 1 x4 + 1 n–1 x4 + 1 Integrals Involving 1. ln ( x ) u 1 3. ∫ x a 2 – x 2 dx = – -- ( a 2 – x 2 ) 3 / 2 3 4. ∫ x 2 a 4 –1 x x a – x dx = -- ( 2 x 2 – a 2 ) a 2 – x 2 + ---- sin -- a 8 8 2 2 a2 – x2 5. ∫ ------------------- dx = x a + a2 – x2 a 2 – x 2 – a ln ---------------------------x –1 x a2 – x2 1 6. ∫ ------------------- dx = – -- a 2 – x 2 – sin -- a x2 x –1 1 x 7. ∫ ------------------- dx = sin -- a a2 – x2 x 8. ∫ ------------------- dx = – a 2 – x 2 a2 – x2 1 2 ax – x 2 8. ∫ -------------------------- dx = – ----------------------ax x 2 ax – x 2 1 x–a 9. ∫ ------------------------------ dx = ----------------------------( 2 ax – x 2 ) 3 / 2 a 2 2 ax – x 2 Miscellaneous Integrals a 2 –1 x 2a + x a+x 1. ∫ x ----------- dx = – -------------- a 2 – x 2 + ---- sin -2 2 a–x a 2. ∫ cos x dx = sin x 2. ∫ cos ax dx ∫ ∫ 2 x ------------- dx = -- ln ( x 3 / 2 + x 3 – a ) 3 x3 – a x ----------- dx = x+a x 2 + ax – a ln ( x + a + x ) ∫ F ( ln x ) dx = ∫ F ( u ) e du, u = ln x 2. ∫ ln ax dx ∫ ln x dx = – x + x ln x 3. ∫ ( ln x ) n dx = x ( ln x ) n – n ∫ ( ln x ) n – 1 dx 1 4. ∫ -- dx = ln x x = – x + x ln ax a x x x 9. ∫ ------------------- dx = – -- a 2 – x 2 + ---- sin -- a 2 2 a2 – x2 2 2 2 2 1 1 - ---------------------------10. ∫ ---------------------- dx = – -- ln a + a – xa x x a 2 – x2 1 1 11. ∫ ------------------------ dx = – ------- a 2 – x 2 a2 x x2 a 2 – x 2 4 3 a - –1 x x 12. ∫ ( a 2 – x 2 ) 3 / 2 dx = – -- ( 2 x 2 – 5 a 2 ) a 2 – x 2 + ------- sin -- a 8 8 –1 3. 4 1 + x2 1 2 x + 1 + x4. ∫ ------------------------------------ dx = ------ ln ---------------------------------1 – x2 2 ( 1 – x2) 1 + x4 7. ∫ x cos x dx = cos x + x sin x cos ax x sin ax 8. ∫ x cos ax dx = -------------- + ---------------a a2 9. ∫ x 2 cos x dx = x 2 sin x + 2 x cos x – 2 sin x 10. ∫ x cos x dx = x sin x – n ∫ x n n n–1 sin x dx 1 5. ∫ ---------- dx = ln ln x x ln x 1 1 n+1 6. ∫ x n ln x dx = ----------- x ln x – ------------------ x n + 1 ( n + 1) 2 n+1 ax e 7. ∫ e ax sin bx dx = --------------- ( a sin bx – b cos bx ) a2 + b 2 e ax 8. ∫ e ax cos bx d x = ---------------2 ( a cos b x – b sin bx ) a2 + b Hyperbolic Functions x x e x + e –sinh x e x – e –Note: sinh x = ---------------- , cosh x = ---------------- , tanh x = -------------2 2 cosh x 1 1 2 1. ∫ sinh x dx = cosh x 2. ∫ sinh x dx = -- sinh 2 x – -- x 4 2 1 1 2 3. ∫ cosh x dx = sinh x 4. ∫ cosh x dx = -- sinh 2 x + -- x 4 2 5. 6. 7. 9. 1 x 13. ∫ ------------------------- dx = -----------------------( a 2 – x2 ) 3 / 2 a 2 a 2 – x2 Integrals Involving x 2 ± a 2 x2 ± a2 DERIVATIVES dw dw du ------ = ------ ----- : Chain Rule -dx du dx d du d ----- f ( u ) = ----- ----- f ( u ) -dx dx du –1 du dx –1 1 ----- = ----- = -------------- du dx dx ⁄ du d du dv ----- ( u + v ) = ----- + ----dx dx dx d- -1 ------------- u = ---- v du – u dv dx dx v v 2 dx 11. 12. ∫ cos ∫ cos –1 x dx = x cos x – 1 – x 2 –1 1 –1 ax dx = x ( cos ax ) – -- 1 – a 2 x 2 a 1 dx 13. ∫ ---------------- = -- tan ax 2 a cos ax –1 –1 x Note: ln x + x 2 + a 2 = sinh -- , a x a a + x2 + a 2 ln x + x 2 – a 2 = cosh -- , ln ----------------------------- = sinh -- a x x –1 d----- ( uv ) = u dv + v du --------dx dx dx dv du d ----- ( u v ) = vu v – 1 ----- + u v ln u ----dx dx dx d 1 ----- ln x = -dx x d ----- a x = a x ln a dx d ( log au ) 1 du --------------------- = ( log ae ) -- ----u dx dx d ( sin x ) ⁄ dx = cos x d ( tan x ) ⁄ dx = sec x d ( sec x ) ⁄ dx = sec x tan x –1 2 sin ( a + b ) x sin ( a – b ) x = --------------------------- + --------------------------2 (a + b ) 2(a – b) 1 1 ax 15. ∫ ----------------------- dx = -- tan ----a 1 + cos ax 2 14. ∫ cos ax cos bx dx ∫ tanh x dx = ln ( cosh x ) ∫ coth x dx = ln sinh x –1 ∫F( ∫F( x 2 + a 2 ) dx = a ∫ F ( a sec u ) sec u du, x = a tan u 2 x 2 – a 2 ) dx = a ∫ F ( a tan u ) sec u tan u du, x = a sec u 1 1 - x–a 2. ∫ --------------- dx = ----- ln ----------x2 – a2 2a x + a TAN 1. 3. 5. sin x tan x = ---------cos x = – ln cos x 1 2. ∫ tan ax dx = – -- ln ( cos ax ) a 4. ∫ tan x dx ∫ tan ∫ tan 2 ∫ sech x dx = tan ( sinh x ) ∫ sech x tanh x dx = – sech x 10. ∫ csch x coth x dx = – csch x 11. 8. ∫ sech x dx 2 = tanh x –1 1 1 x 1. ∫ --------------- dx = -- tan -- a x 2 + a2 a d ----- e x = e x dx d du ----- ( u n ) = nu n – 1 ----dx dx x dx = tan x – x ∫ tan 2 1 ax dx = – x + -- tan ax a 1 n–1 n–2 x dx = ----------- tan x – ∫ tan x dx n–1 1 –1 –1 6. ∫ tan x dx = x tan x – -- ln ( x 2 + 1 ) 2 n 1 = ln tanh -- x 2 ax e 12. ∫ e ax sinh bx dx = --------------- ( a sinh bx – b cosh bx ) a2 – b2 eax ax cosh bx dx = --------------- ( a cosh bx – b sinh bx ) 13. ∫ e a2 – b2 ∫ csch x dx x 1 3. ∫ --------------- dx = -- ln x 2 + a 2 x 2 + a2 2 a2 x 2 2 4. ∫ x ± a dx = -- x 2 ± a 2 ± ---- ln x + x 2 ± a 2 2 2 12 5. ∫ x x 2 ± a 2 dx = -- ( x ± a 2 ) 3 / 2 3 a4 x 6. ∫ x 2 x 2 ± a 2 dx = -- ( 2 x 2 ± a 2 ) x 2 ± a 2 – ---- ln x + x 2 ± a 2 8 8 x2 + a2 7. ∫ ------------------- dx = x x2 – a2 8. ∫ ------------------- dx = x a x 2 + a 2 – a sinh -- x –1 x x 2 – a 2 – a sec -a –1 d ( cos x ) ⁄ dx = – sin x d ( cot x ) ⁄ dx = – csc x d ( csc x ) ⁄ dx = – csc x cot x 2 d ( sin u ) π 1 du π ----------------------- = ----------------- ----- for – -- ≤ sin –1u ≤ -- 2 2 dx 1 – u 2 dx – 1 du d ( cos u ) ----------------------- = ----------------- ----- for ( 0 ≤ cos – 1u ≤ π ) -dx 1 – u 2 dx d ( tan u ) 1 du π π ---------------------- = ------------- ----- for – -- < tan –1u < -- -2 dx 1 + u 2 dx 2 – 1 du d ( cot u ) ---------------------- = ------------- ----- for ( 0 < cot –1u < π ) -dx 1 + u 2 dx d ( sec u ) π 1 - ---------------------------- = -------------------- du 0 ≤ sec – 1u < π – π ≤ sec – 1u < – -- 2 2 dx u u 2 – 1 dx d ( csc u ) – 1 - ----------------------------- = -------------------- du 0 < csc – 1u ≤ π – π < csc – 1u ≤ – π 2 2 dx u u 2 – 1 dx d ( sinh x ) ⁄ dx = cosh x d ( cosh x ) ⁄ dx = sinh x d ( tanh x ) ⁄ dx = sech x d ( coth x ) ⁄ dx = – csch x d ( sech x ) ⁄ dx = – sech x tanh x d ( csch x ) ⁄ dx = – csch x coth x 1 du d ----- sinh –1u = ----------------- -----dx u 2 + 1 dx 1 du d ----- cosh –1u = ----------------- ----- for u > 1 and cosh –1u ≥ 0 -dx u 2 – 1 dx 1 du d ------ tanh –1u = ------------- ----- for – 1 < u < 1 1 – u 2 dx dx 2 2 –1 –1 –1 –1 –1 1 1 1 7. ∫ ----------------------- dx = -- x + -- ln ( cos ax + sin ax ) 1 + tan ax 2 a COT 1. 3. 5. cos x 1 cot x = ---------- = ---------sin x tan x = ln sin x = – cot x – x 1 2. ∫ cot ax dx = -- ln ( sin ax ) a 1 2 4. ∫ cot ax dx = – x – -- cot ax a ∫ cot x dx ∫ cot x dx 2 1 n n–1 n–2 ∫ cot x dx = – ----------- cot x – ∫ cot x dx n–1 1 –1 –1 -- ln ( x 2 + 1 ) 6. ∫ cot x dx = x cot x + 2 SEC 1. 2. 3. 4. 1 sec x = ---------cos x = ln sec x + tan x Integrals Involving a + bx 1 ∫ F ( a + bx ) dx = -- ∫ F ( u ) du, u = a + bx b 1 1 1. ∫ -------------- dx = -- ln a + bx b a + bx 1 –1 2. ∫ --------------------- dx = ----------------------( a + bx ) 2 b ( a + bx ) –1 1 --------------------- dx = -----------------------------------------------1 ( n ≠ 1 ) 3. ∫ ( n – 1 ) b ( a + bx ) n – ( a + bx ) n 1 x 4. ∫ -------------- dx = ---- [ a + bx – a ln a + bx ] b2 a + bx a x 1 5. ∫ --------------------- dx = ---- -------------- + ln a + bx ( a + bx ) 2 b 2 a + bx xa + 2 bx 6. ∫ --------------------- dx = – -----------------------------( a + bx ) 3 2 b 2 ( a + bx ) 2 11 x2 -7. ∫ -------------- dx = ---- -- ( a + bx ) 2 – 2 a ( a + bx ) + a 2 ln a + bx b3 2 a + bx a2 x2 18. ∫ --------------------- dx = ---- a + bx – -------------- – 2 a ln a + bx b3 ( a + bx ) 2 a + bx ( a + bx ) n + 1 9. ∫ ( a + bx ) n dx = --------------------------- ( n ≠ – 1 ) b ( n + 1) 1 1 x10. ∫ ---------------------- dx = -- ln -------------a a + bx x ( a + bx ) 1 1 b - -------------11. ∫ ------------------------ dx = – ----- + ---- ln a + bx x 2 ( a + bx ) ax a 2 x 1 1 1 x 12. ∫ ------------------------2 dx = ----------------------- + ---- ln -------------x ( a + bx ) a ( a + bx ) a 2 a + bx a + bx Integrals Involving 2 ∫ F ( a + bx ) dx = -- ∫ uF ( u ) du, u = a + bx b n ∫ F ( n a + bx ) dx = -- ∫ u n – 1 F ( u ) du, u = n a + bx b x2 ± a2 x2 ± a 2 9. ∫ ------------------- dx = – ------------------- + ln x + x 2 ± a 2 x2 x 1 10. ∫ ------------------- dx = ln x + x 2 ± a 2 x2 ± a2 2 2 1 − x ± a11. ∫ ------------------------ dx = + ------------------a2x x2 x 2 ± a 2 ∫ sec x dx ∫ sec x dx 2 1 ∫ sec ax dx = -- ln ( sec ax + tan ax ) a = tan x 1 1 = -- sec x tan x + -- ln sec x + tan x 2 2 n–2 1 n n–2 n–2 5. ∫ sec x dx = ----------- sec x tan x + ----------- ∫ sec x dx n–1 n–1 ∫ sec x dx 3 1 1 a 12. ∫ ---------------------- dx = – -- sinh -- x a x x2 + a 2 –1 1 1 x 13. ∫ ---------------------- dx = -- sec -a a x x2 – a2 x 2 ± a2 14. ∫ ------------------- dx = x x2 ± a2 2 x2 x − a15. ∫ ------------------- dx = -- x 2 ± a 2 + ---- ln x + x 2 ± a 2 2 2 x2 ± a2 16. 3 a4 x ∫ ( x 2 ± a 2 ) 3 / 2 dx = -- ( 2 x 2 ± 5 a2 ) x 2 ± a2 + -------- ln x + x 2 ± a 2 8 8 1 x 17. ∫ -------------------------2 dx = ± ------------------------( x2 ± a2 ) 3 / a2 x2 ± a2 Integrals Involving ax 2 + bx + c 1 1 a 1. ∫ ---------------- dx = --------- tan x -- c ax 2 + c ac –1 1 2 2 ax + b ----------------------------- dx = ------------------------ tan ------------------------ 2a. ∫ 2 4 ac – b 2 ax + bx + c 4 ac – b 2 for 4 ac – b 2 > 0 1 1 2 ax + b – b 2 – 4 ac 2b. ∫ ----------------------------- dx = ------------------------ ln ------------------------------------------------ax 2 + bx + c b 2 – 4 ac 2 ax + b + b 2 – 4 ac for b 2 – 4 ac > 0 –1 –1 6. ∫ sec –1 x dx = x sec x – ln x + x 2 – 1 –1 CSC 1. 2. 3. 1 csc x = --------sin x = – ln csc x + cot x 1 = -- ln ( csc ax – cot ax ) a = – cot x ∫ csc x dx ∫ csc x dx 2 ∫ csc ax dx ...
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This note was uploaded on 03/09/2009 for the course MATH 101 taught by Professor Spencer during the Spring '09 term at CUNY City.

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