NAME____________________________________ SCHOOL__________________________________ TEST 2 Monday, February 25, 2008 Fax your answers to: Irvin Roy Hentzel at Fax 515-294-5454 Snail Mail your answers to: Irvin Roy Hentzel Department of Mathematics 432 Carver Hall Iowa State University Ames, Iowa 50011-2064 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 -Ln(Cos[u])+C ----------------------------------- du (g) INT ---------------- 2 2 Sqrt[ a - u ] ArcSin[u/a] + C ----------------------------------- r r+1 (h) INT u du u /(r+1) + C ----------------------------------- u u (i) INT a du a /ln(a) + C ----------------------------------- (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 Ln(Sin[u]) + C ----------------------------------- du (n) INT ----------- 2 2 a + u (1/a) ArcTan[u/a] + C ----------------------------------- du (o) INT ---------------- 2 2 u Sqrt[ u - a ] (1/a) ArcSec[u/a] + C -------------------------------- (p) INT Sinh[u] du Cosh[u] + C ----------------------------------- (q) INT Cosh[u] du Sinh[u] + C ----------------------------------- (r) INT Sec[u] du Ln( Sec[u] + Tan[u] ) + C ----------------------------------- (s) INT Csc[u] du -Ln(Csc[u] + Ctn[u]) + C ----------------------------------- In problems 2 through 9 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 substitute in the appropriate limits. 2 x 2. Find INT ----------- dx 2 3 Cos [x ] 3 u = x (a) ____________________________________________ 2 INT 1/3 Sec [u] du (b) ____________________________________________ 1/3 Tan[u] + C (c) ____________________________________________ 3 1/3 Tan[x ] + C (d) ____________________________________________ 5 3. Find INT --------------- dx 2 Sqrt[ 9 - 4x ] u = 2x (a) ____________________________________________ 5/2 INT ------------ du 2 Sqrt[9-u ] (b) ____________________________________________ 5/2 ArcSin[u/3] + C (c) ____________________________________________ 5/2 ArcSin[2x/3] + C (d) ____________________________________________ _ _ | | | 3 | | x | 2 |_ _| 4. Find INT x e dx 3 u = x (a) ____________________________________________ u INT 1/3 e du (b) ____________________________________________ u 1/3 e + C (c) ____________________________________________ 3 x 1/3 e + C (d) ____________________________________________ Sec[x] Tan[x] 5. Find INT ------------------- dx 2 1 + Sec [x] u = Sec[x] (a) ____________________________________________ du INT ---------- 2 1+u (b) ____________________________________________ ArcTan[u] + C (c) ____________________________________________ ArcTan[Sec[x]] + C (d) ____________________________________________ 6. Find INT (1/x) Cos[ Ln(x) ] dx u = Ln(x) (a) ____________________________________________ INT Cos[u] du (b) ____________________________________________ Sin[u] + C (c) ____________________________________________ Sin[ Ln(x) ] + C (d) ____________________________________________ Sec[t] a 7. Find INT ------------- dt Cos[t] Ctn[t] u = Sec[t] (a) ____________________________________________ u INT a du (b) ____________________________________________ u a / Ln(a) + C (c) ____________________________________________ Sec[t] a / Ln(a) + C (d) ____________________________________________ t=5 2 8. Evaluate INT t Sqrt[ t - 4 ] dt t=2 2 u = t - 4 (a) ____________________________________________ t=5 1/2 INT 1/2 u du t=2 (b) ____________________________________________ | 3/2 | t=5 (1/2) | u | |-------- | | 3/2 | t=2 (c) ____________________________________________ 3/2 | u = 21 3/2 1/3 u | = 1/3 21 | u = 0 (d) ____________________________________________ x=3 2 3 9. Find INT x Sqrt[ x + 11 ] dx x=1 3 u = x + 11 (a) ____________________________________________ 1/2 INT 1/3 u du (b) ____________________________________________ 3/2 | x=3 1/3 u | ------- | 3/2 | x=1 (c) ____________________________________________ 3/2 | u = 38 3/2 3/2 2/9 u | = 2/9 (38 - 12 ) | u = 12 (d) ____________________________________________ In problems 10 through 12 use integration by parts. Set up the array u v each time you do integration by parts. du dv 2 10. Find INT x Sec [x] dx x Tan[x] 2 dx Sec x dx x Tan[x] - INT Tan[x] dx x Tan[x] + Ln(Cos[x]) + C <==================== 8 11. Find INT x Ln(x) dx 9 Ln(x) x /9 8 1/x dx x dx 9 8 x Ln(x)/9 - 1/9 INT x dx 9 9 x Ln(x)/9 - 1/81 x + C <========================== 12. Find INT ArcTan[x] dx ArcTan[x] x dx --------- dx 2 1+x x dx x ArcTan[x] - INT-------- 2 1+x 2 x ArcTan[x] -1/2 Ln(1+x ) + C <======================= 3 4/3 13. Find INT Sin [x] Cos [x] dx 2 4/3 INT Sin [x] Cos [x] Sin[x] dx 2 4/3 INT (1-Cos [x] ) Cos [x] Sin[x] dx 4/3 10/3 INT (Cos [x] - Cos [x] ) Sin[x] dx 7/3 13/3 -Cos [x] Cos [x] -------------- + ------------ + C <======================= 7/3 13/3 -7/2 2 14. Find INT Tan [x] Sec [x] dx -5/2 Tan [x] -------------- + C <======================= -5/2 3 5/7 15. Find INT Tan [x] Sec [x] dx 2 5/7 INT Tan [x] Sec [x] Tan[x] dx 2 5/7 INT (Sec [x] -1) Sec [x] Tan[x] dx 19/7 5/7 INT (Sec [x] - Sec [x] ) Tan[x] dx 12/7 -2/7 INT (Sec [x] - Sec [x] ) Sec[x] Tan[x] dx 19/7 5/7 Sec [x] Sec [x] ------------ - --------- + C <===================== 19/7 5/7 -