Friday, February 22 Practice test. The real test is Monday, February 25, 2008 1. Give the standard Integral forms (a) INT k du k u + C u u (b) INT e du e + C (c) INT Sin[u] du -Cos[u] + C 2 (d) INT Sec [u] du Tan[u] + C (e) INT Sec[u] Tan[u] du Sec[u] + C (f) INT Tan[u] du -Log[Cos[u]] + C du (g) INT ---------------- ArcSin[u/a] + C 2 2 Sqrt[ a - u ] r r+1 (h) INT u du u --------- + C r+1 u u (i) INT a du a ------- + C Log[a] (j) INT Cos[u] du Sin[u] + C 2 (k) INT Csc [u] du -Ctn[u] + C (l) INT Csc[u] Ctn[u] du -Csc[u] + C (m) INT Ctn[u] du Log[ Sin[u] + C du (n) INT ----------- 1/a Arc Tan[u/a] + C 2 2 a + u du (o) INT ---------------- 1/a ArcSec[u/a] + C 2 2 u Sqrt[ u - a ] (p) INT Sinh[u] du Cosh[u] + C (q) INT Cosh[u] du Sinh[u] + C (r) INT Sec[u] du Log[ Sec[u]+Tan[u] ] + C (s) INT Csc[u] du -Log[ Csc[u] + Ctn[u] ] + C In problems 1 through 8 use substitution. (a) Give the substitution. (b) Rewrite the integral after making the substitution. (c) Integrate the New integral. (d) In the indefinite integral change the substituted variable back to the original variable. In the definite integral subsitute in the appropriate limits. x 1. Find INT ----------- dx 2 2 Cos [x ] 2 u = x (a) ____________________________________________ 1/2 du INT ----------- 2 Cos [u] (b) ____________________________________________ 1/2 Tan[u] + C (c) ____________________________________________ 2 1/2 Tan[ x ] + C (d) ____________________________________________ 3 2. Find INT --------------- dx 2 Sqrt[ 5 - 9x ] u = 3 x (a) ____________________________________________ du INT ------------------ 2 Sqrt[ 5-u ] (b) ____________________________________________ ArcSin[ u/Sqrt[5] ] + C (c) ____________________________________________ ArcSin[ 3x/Sqrt[5] ] + C (d) ____________________________________________ 1/x 6 e 3. Find INT -------- dx 2 x u = 1/x (a) ____________________________________________ u INT -6 e du (b) ____________________________________________ u -6 e + C (c) ____________________________________________ 1/x -6 e + C (d) ____________________________________________ x e 4. Find INT ---------------- dx 2x 4 + 9 e x u = 3 e (a) ____________________________________________ 1/3 du INT --------- 2 4 + u (b) ____________________________________________ 1/6 ArcTan[u/2] + C (c) ____________________________________________ x 1/6 ArcTan[ 3 e /2 ] + C (d) ____________________________________________ 2 5. Find INT x Cos[x ] dx 2 u = x (a) ____________________________________________ INT 1/2 Cos[u] du (b) ____________________________________________ 1/2 Sin[u] + C (c) ____________________________________________ 2 1/2 Sin[x ] + C (d) ____________________________________________ Page 385 Tan[t] a 6. Find INT ----------- dt 2 Cos [t] u = Tan[t] (a) ____________________________________________ u INT a du (b) ____________________________________________ u a ------ + C Log[a] (c) ____________________________________________ Tan[t] a ---------- + C Log[a] (d) ____________________________________________ Page 385 x=5 2 7. Evaluate INT t Sqrt[ t - 4 ] dt x=2 2 u = t - 4 (a) ____________________________________________ x=5 1/2 INT 1/2 u du x=2 (b) ____________________________________________ 3/2 | x=5 1/2 u | --------- | 3/2 | x=2 (c) ____________________________________________ 3/2 | u = 21 3/2 1/3 u | = 1/3 21 | u = 0 (d) ____________________________________________ x=3 3 4 8. Find INT x Sqrt[ x + 11 ] dx x=1 4 u = x + 11 (a) ____________________________________________ 1/2 INT 1/4 u du (b) ____________________________________________ 3/2 | x=3 u | 1/4 -------- | 3/2 | x=1 (c) ____________________________________________ 3/2 | u=92 3/2 3/2 1/6 u | 1/6( 92 - 12 ) | u=12 (d) ____________________________________________ In problems 9 through 14 use integration by parts. Set up the array u v each time you do integration by parts. du dv 9. Find INT x Cos[x] dx x Sin[x] dx Cos[x] dx x Sin[x] - INT Sin[x] dx x Sin[x] + Cos[x] <==================== Page 388 x=2 10. Find INT ln(x) dx x=1 ln(x) x 1/x dx dx x ln(x) - INT dx | x=2 x ln(x) - x | = (2 ln(2) - 2 -(0-1) | x=1 2 ln(2) -1 <=========================== 11. Find INT ArcSin[x] dx ArcSin[x] x dx -------------- dx 2 Sqrt[1-x ] -x dx = x ArcSin[x] + INT --------------- 2 Sqrt[1-x ] 2 1/2 (1-x ) = x ArcSin[x] - ----------- + C <======================= 1/2 (-2) t=2 6 12. Find INT t ln(t) dt t=1 7 t ln(t) ------- 7 1/t dt 6 t dt 7 6 t ln(t) t ------------- - INT ----- dt 7 7 7 7 | t=2 t ln(t) t | ------------- - ------ | 7 49 | t=1 128 Log[2]/7 - 128/49 + 1/49 = 128 Log[2]/7 - 127/49 2 13. Find INT x Sin[x] dx 2 + x Sin[x] - 2 x -Cos[x] + 2 -Sin[x] - 0 Cos[x] 2 -x Cos[x] +2 x Sin[x] +2 Cos[x] + C x 14. Find INT e Sin[x] dx x e -Cos[x] x e dx Sin[x] dx x x x INT e Sin[x] dx = -e Cos[x] + INT e Cos[x] dx x e Sin[x] x e dx Cos[x] dx x x x x INT e Sin[x] dx = -e Cos[x] + e Sin[x] - INT e Sin[x] dx x x x 2 INT e Sin[x] dx = -e Cos[x] + e Sin[x] x x e (-Cos[x] + Sin[x]) INT e Sin[x] dx = ------------------- <============ 2 Page 390 n 15. Derive a reduction formula for INT Sin [x] dx n-1 Sin [x] -Cos[x] n-2 (n-1) Sin [x] Cos[x] dx Sin[x] dx n n-1 n-2 2 INT Sin [x] dx = -Sin [x] Cos[x] + (n-1) INT Sin [x] Cos [x] dx n n-1 n-2 2 INT Sin [x] dx = -Sin [x] Cos[x] + (n-1) INT Sin [x] (1-Sin [x]) dx n n-1 n-2 INT Sin [x] dx = -Sin [x] Cos[x] + (n-1) INT Sin [x] dx n -(n-1) INT Sin [x] dx n n-1 n-2 n INT Sin [x] dx = -Sin [x] Cos[x] + (n-1) INT Sin [x] dx n-1 n -Sin [x] Cos[x] (n-1) n-2 INT Sin [x] dx = -------------------- + ----- INT Sin [x] dx <======= n n x=Pi/2 8 16. Use the reduction formula to above to evaluate INT Sin [x] dx x=0 x=Pi/2 8 7 | x=Pi/2 INT Sin [x] dx = -Sin [x] Cos[x] | x=0 ---------------- | 8 | | 5 | -7/8 1/5 Sin [x] Cos[x] | | | 3 | -7/8 5/6 1/4 Sin [x] Cos[x] | | | | -7/8 5/6 3/4 1/2 Sin[x] Cos[x] |+ | 7/8 5/6 3/4 1/2 x | x=0 7/8 5/6 3/4 1/2 Pi/2 = 35 Pi/256 Page 394 5 17. Find INT Sin [x] dx 4 INT Sin [x] Sin[x] dx 2 2 INT (1 - Cos [x]) Sin[x] dx 2 4 INT (1 - 2 Cos [x] + Cos [x]) Sin[x] dx 3 5 Cos [x] Cos [x] -Cos[x] +2 ------- - --------- + C 3 5 In problems 18 through 26 use the techniques for integrating powers of signs and cosines, secants, tangents, and the other trigonemetric functions. Show your work. 2 18. Find INT Sin [x] dx 1-Cos[2x] INT --------- dx 2 x/2 - Sin[2x]/4 + C <=============== 4 19. Find INT Cos [x] dx 1+Cos[2x] 2 INT (---------- ) dx 2 2 1/4 INT 1 + 2 Cos[2x] + Cos [2x] dx 1+Cos[4x] 1/4 INT 1 + 2 Cos[2x] + --------- dx 2 1/4( x + Sin[2x] + x/2 + Sin[4x]/8 ) + C <============ 3 -4 20. Find INT Sin [x] Cos [x] dx 2 -4 INT (1-Cos [x]) Cos [x] Sin[x] dx -4 -2 INT (Cos [x] - Cos [x] ) dx -3 -1 Cos [x] Cos [x] - -------- + ----------- + C -3 -1 2 4 21. Find INT Sin [x] Cos [x] dx 1-Cos[2x] 1+Cos[2x] 1+Cos[2x] INT --------- --------- ---------- dx 2 2 2 2 1-Cos [2x] 1+Cos[2x] INT --------- --------- dx 4 2 2 3 1/8 INT 1 + Cos[2x] - Cos [2x] - Cos [2x] dx 1+Cos[4x] 2 1/8 INT 1 + Cos[2x] - (---------) - (1-Sin [2x]) Cos[2x] dx 2 3 1/8 ( x + (1/2) Sin[2x] - x/2 - 1/8 Sin[4x] - 1/2 Sin[2x] + 1/6 Sin [2x]) + C 3 1/8 ( x/2 + (1/2) Sin[2x] - 1/8 Sin[4x] -1/2 Sin[2x] + 1/6 Sin [2x]) + C <=== Page 395 22. Find INT Sin[2x] Cos[3x] dx Sin[2x] 1/3 Sin[3x] 2 Cos[2x] dx Cos[3x] dx INT Sin[2x] Cos[3x] dx = 1/3 Sin[2x] Sin[3x] - 2/3 INT Cos[2x] Sin[3x] dx Cos[2x] -1/3 Cos[3x] -2 Sin[2x] dx Sin[3x] dx INT Sin[2x] Cos[3x] dx = 1/3 Sin[2x] Sin[3x] - 2/3 ( -1/3 Cos[2x] Cos[3x] -2/3 INT Sin[2x] Cos[3x]) (1-4/9) INT Sin[2x] Cos[3x] dx = 1/3 Sin[2x] Sin[3x] +2/9 Cos[2x] Cos[3x] + C INT Sin[2x] Cos[3x] = 3/5 Sin[2x] Sin[3x] + 2/5 Cos[2x] Cos[3x] + C <======= Page 397 4 23. Find INT Ctn [x] dx 2 2 INT Ctn [x] (Csc [x]-1) dx 2 2 2 INT Ctn [x] Csc [x] dx - INT Ctn [x] dx 2 2 2 INT Ctn [x] Csc [x] dx - INT Csc [x]-1 dx 3 -Ctn [x] ------------- + Ctn[x] + x + C 3 5 24. Find INT Tan [x] dx 2 2 INT (Sec [x] -1) Tan[x] dx 4 2 INT ( Sec [x] - 2 Sec [x] + 1) Tan[x] dx 3 INT ( Sec [x] - 2 Sec[x] ) Sec[x] Tan[x] dx + INT Tan[x] dx 4 2 Sec [x] Sec [x] --------- - 2 ------ - Log[ Cos[x] ] + C 4 2 Page 398 -3/2 4 25. Find INT Tan [x] Sec [x] dx -3/2 2 2 INT Tan [x] (Tan [x] + 1) Sec [x] dx 1/2 -3/2 2 INT ( Tan [x] + Tan [x] ) Sec [x] dx 3/2 -1/2 Tan [x] Tan [x] ---------- + ------------ + C 3/2 -1/2 3 -1/2 26. Find INT Tan [x] Sec [x] dx 2 -1/2 INT (Sec [x] -1) Sec [x] Tan[x] dx 1/2 -3/2 INT (Sec [x] - Sec [x] ) Sec[x] Tan[x] dx 3/2 -1/2 Sec [x] Sec [x] ----------- - ---------- + C 3/2 -1/2 In problems 27 through 29, make a rationalizing substitution. a) Give the rationalizing substitution b) Give dx c) Make the substitution d) Integrate e) Substitute back the original unknowns dx 27. Find INT ------------ x - Sqrt[x] u = Sqrt[x] 2 u = x 2 u du = dx 2 u du INT ----------- 2 u - u du 2 INT -------- u-1 2 Log[u-1] + C 2 Log[Sqrt[x]-1] + C _____ 3/ 28. Find INT x \/ x-4 dx ______ 3/ u = \/ x-4 3 u = x-4 2 3 u du = dx 3 2 INT (u + 4) u 3 u du 6 3 3 INT u + 4 u du 7 u 4 3( ---- + u ) + C 7 7/3 4/3 3/7 (x-4) + 3 (x-4) + C ________ 5/ 2 29. Find INT x \/ (x+1) dx 1/5 u = (x+1) 5 u = x+1 4 5 u du = dx 5 2 4 INT (u - 1) u 5 u du 11 6 5 INT u - u du 12 7 5 u 5 u ----- - ----- + C 12 7 12/5 7/5 5 (x+1) 5 (x+1) --------- - -------- + C 12 7 2 2 30. Find INT Sqrt[ a - x ] dx | | /| 2 2 | a / | Sqrt[a - x ] | / | | / | |/t | ---------------+----------------------------- | x | | | x = a Cos[t] dx = -a Sin[t] 2 2 Sqrt[a - x ] = a Sin[t] INT a Sin[t] (-a Sin[t]) dt 2 2 -a INT Sin [t] dt 2 1-Cos[2t] -a INT ---------- dt 2 2 -a ( t/2 - Sin[2t]/4) ) + C 2 -1/2 a t +1/2 a Sin[t] a Cos[t] + C 2 2 2 -1/2 a ArcCos[x/a] +1/2 x Sqrt[a - x ] + C -1/2 a^2 ArcCos[x/a] + 1/2 x Sqrt[a^2-x^2] <================ In problems 31 through 34, a) draw the triangle and label the sides. b) Find dx c) Make the substitution d) Integrate e) Substitute back the original unknowns. Page 401 dx 31 . Find INT ------------ 2 Sqrt[ 9 + x ] | | 2 /| Sqrt[9+x ] / | | / | | / | 3 | / | | / | |/t | --------+---------------------------- | x | x = 3 Ctn[t] 2 dx = -3 Csc [t] 2 Sqrt[9+x ] = 3 Csc[t] 2 -3 Csc [t] dt INT ------------------ = INT -Csc[t] dt 3 Csc[t] = Log[ Csc[t] + Ctn[t] ] + C = Log[ 1/3 Sqrt[9+x^2] + x/3] + C <============== Page 402 2 x=4 Sqrt[ x - 4 ] 32. Calculate INT -------------- dx x=2 x /| | / | | / | | x/ | 2 | / | Sqrt[x -4] | / | |/t | --------+----------------------- | 2 | | x = 2 Sec[t] dx = 2 Sec[t] Tan[t] dt 2 Sqrt[x -4] = 2 Tan[t] x=4 2 Tan[t] INT ------------- 2 Sec[t] Tan[t] dt x=2 2 Sec[t] x=4 2 INT 2 Tan [t] dt x=2 x=4 2 2 INT Sec [t]-1 dt x=2 t=Pi/3 | | x=4 2| Tan[t] - t | | | x=2 t=0 2( Tan[Pi/3] - Pi/3) = 2( Sqrt[3]-Pi/3 ) <==================== dx 33. Find INT ---------------- 2 Sqrt[ x + 2 x + 26 ] dx Find INT ---------------- 2 Sqrt[ (x+1) + 25 ] 2| /| Sqrt[(x+1) +25] / | | / | | / |5 | / | | / | |/t | -------+----------------------------- | x+1 | x+1 = 5 Ctn[t] 2 dx = -5 Csc [t] dt 2 Sqrt[ (x+1) + 25 ] = 5 Csc[t] 2 -5 Csc [t] dt INT ----------------- 5 Csc[t] INT - Csc[t] dt Log[ Csc[t] + Ctn[t] ] + C Log[ 1/5 Sqrt[x^2 + 2x + 26] + (x+1)/5 ] + C <=========== 2x 34. Find INT -------------------- dx 2 Sqrt[ x + 2 x + 26 ] 2| /| Sqrt[(x+1) +25] / | | / | | / |5 | / | | / | |/t | -------+----------------------------- | x+1 x+1 = 5 Ctn[t] 2 dx = -5 Csc [t] dt 2 Sqrt[ (x+1) + 25 ] = 5 Csc[t] 2(5 Ctn[t]-1) 2 INT -------------- (-5 Csc [t] dt) 5 Csc[t] -2 INT ( 5 Ctn[t]-1) Csc[t] dt -10 INT Csc[t] Ctn[t] + 2 INT Csc[t] 10 Csc[t] - 2 Log[ Csc[t] + Ctn[t] ] + C 10(1/5) Sqrt[x^2+2x+26] -2 Log[ 1/5 Sqrt[x^2+2x+26] + (x+1)/5 ] + C 2 35. Find INT -------- dx 3 (x+1) -2 2 (x+1) ------------ + C -2 In problems 36 through 42, set up the partial fraction decomposition, Evaluate the unknown cofficients and integrate. Page 404 2x + 2 36. Find INT ------------- dx 2 x - 4 x + 8 This is already in the partial fraction decomposition. It is ready to integrate. 2x -4 +6 INT ------------ dx + INT ------------------ dx 2 2 x - 4 x + 8 x - 4 x + 8 2 dx Log[ x - 4 x + 8 ] + 6 INT --------------- 2 (x-2) + 4 2 Log[ x - 4 x + 8 ] + 6/2 ArcTan[(x-2)/2] + C 3x-1 37. Find INT ----------- dx 2 x - x - 6 (-2) (3) -7 8 ------- ------ 3x - 1 -5 5 ------------- = ------------- + --------------- (x+2)(x-3) x+2 x-3 / g (x) = 2x - 1 = 7/5 Log[x+2] + 8/5 Log[x-3] + C <================ 5x + 3 38. Find INT ---------------- dx 3 2 x - 2 x - 3 x (0) (-1) (3) 3 -2 18 ----- ------- ------- -3 4 12 5x+3 ---------------- = -------------- + ----------- + ----------- x(x + 1)( x - 3) x x+1 x-3 / 2 g (x) = 3x -4x -3 = -Log[x] -1/2 Log[x+1] + 3/2 Log[x-3] + C <======== x 39. Find INT --------- dx 2 (x-3) x A B ---------- = ------ + ----- 2 2 (x-3) (x-3) (x-3) x = A (x-3) + B A = 1 B = 3 x 1 3 ---------- = ------ + ----- 2 2 (x-3) (x-3) (x-3) -1 ANS = Log[x-3] -3(x-3) + C 2 3 x - 8 x + 13 40. Find INT -------------------- dx 2 (x+3)(x-1) 2 3 x - 8 x + 13 A B C ------------------ = -------------- + -------- + -------- 2 2 (x+3)(x-1) x+3 x-1 (x-1) 2 2 3 x - 8 x + 13 = A(x-1) + B(x+3)(x-1) + C(x+3) 2 2 = A( x - 2 x + 1) + B(x +2 x - 3) + C(x+3) A + B = 3 -2 A + 2 B + C = -8 A - 3 B + 3 C = 13 1 1 0 3 -2 2 1 -8 1 -3 3 13 1 1 0 3 0 4 1 -2 0 -4 3 10 1 1 0 3 0 4 1 -2 0 0 4 8 1 1 0 3 0 4 1 -2 0 0 1 2 1 1 0 3 0 4 0 -4 0 0 1 2 1 1 0 3 0 1 0 -1 0 0 1 2 1 0 0 4 0 1 0 -1 0 0 1 2 2 3 x - 8 x + 13 4 -1 2 INT ---------------- dx = INT ---------- + -------- + -------- dx 2 2 (x+3)(x-1) x+3 x-1 (x-1) = 4 Log[x+3] - Log[x-1] -2/(x-1) + C 2 6 x - 3 x + 1 41. Find INT ---------------- dx 2 (4x+1)(x + 1) 2 6 x - 3 x + 1 A B x + C -------------------- = ------------ + ------------ 2 2 (4 x + 1)(x + 1) 4 x + 1 x + 1 2 2 6 x - 3 x + 1 = A ( x + 1 ) + ( 4 x + 1 )( B x + C) 2 2 = A (x + 1 ) + B( 4 x + x) + C( 4 x + 1) A + 4 B = 6 -4 B + 4 C = -3 -1 A + C = 1 -17 C = 17 so C = -1 A = 2 B = 1 2 6 x - 3 x + 1 2 x - 1 INT ------------------ dx = INT ------------ + ------------ dx 2 2 (4 x + 1)(x + 1) 4 x + 1 x + 1 2 6 x - 3 x + 1 2 x - 1 INT ------------------ dx = INT ------------ + -------- + ------- dx 2 2 2 (4 x + 1)(x + 1) 4 x + 1 x + 1 x + 1 = 1/2 Log[4x+1] + 1/2 Log[x^2 + 1] - ArcTan[x] + C <===== 2 6 x - 15 x + 22 42. Find INT -------------------- dx 2 2 (x+3)(x + 2 ) 2 6 x - 15 x + 22 A B x + C D x + E -------------------- = -------- + ---------- + ------------- 2 2 2 2 2 (x+3)(x + 2 ) (x+3) x + 2 (x + 2) 2 2 2 2 6 x - 15 x + 22 = A ( x + 2 ) + (B x + C)(x+3)(x + 2) + (D x + E)(x+3) 4 2 3 2 A( x + 4 x + 4 ) + (B x + C)( x + 3 x + 2 x + 6 ) + (D x + E)(x+3) 4 2 A( x + 4 x + 4 ) 4 3 2 B( x + 3 x + 2 x + 6 x ) 3 2 C( x + 3 x + 2 x + 6 ) 2 D( x + 3 x ) E( x + 3 ) A B C D E 1 1 0 0 0 0 0 3 1 0 0 0 4 2 3 1 0 6 0 6 2 3 1 -15 4 0 6 0 3 22 1 1 0 0 0 0 0 3 1 0 0 0 0 -2 3 1 0 6 0 6 2 3 1 -15 0 -4 6 0 3 22 1 1 0 0 0 0 0 3 1 0 0 0 0 -2 3 1 0 6 0 6 2 3 1 -15 0 -22 0 -9 0 67 1 1 0 0 0 0 0 3 1 0 0 0 0 -2 3 1 0 6 0 12 -7 0 1 -33 0 -40 27 0 0 121 1 1 0 0 0 0 0 3 1 0 0 0 0 -11 0 1 0 6 0 33 0 0 1 -33 0 -121 0 0 0 121 1 1 0 0 0 0 0 3 1 0 0 0 0 -11 0 1 0 6 0 33 0 0 1 -33 0 1 0 0 0 -1 1 0 0 0 0 1 0 0 1 0 0 3 0 0 0 1 0 -5 0 0 0 0 1 0 0 1 0 0 0 -1 1 0 0 0 0 1 0 1 0 0 0 -1 0 0 1 0 0 3 0 0 0 1 0 -5 0 0 0 0 1 0 2 6 x - 15 x + 22 1 - x + 3 -5 x -------------------- = -------- + ---------- + ------------- 2 2 2 2 2 (x+3)(x + 2 ) (x+3) x + 2 (x + 2) 2 6 x - 15 x + 22 1 - x + 3 -5 x INT ---------------- dx = INT -------- + ------ + ----- + ------------- 2 2 2 2 2 2 (x+3)(x + 2 ) (x+3) x + 2 x + 2 (x + 2) = Log[x+3] -1/2 Log[x^2+2] + 3/Sqrt[2] ArcTan[x/Sqrt[2]] +5/2 (x^2+2)^(-1) + C