Question

Evaluate the following integrals (from A to E) A. Integration by parts i) ſ (3+ ++2) sin(2t) dt ii) Z dz un (ricos x?cos 4x dx wja iv) (2 + 5x)eš dr. B. Involving Trigonometric functions 271 п i) | sin? ({x)cos*(xx) dx ii) Sco -> (=w) sins (įw) iii) sec iv) ſ tan” (63)sec^® (6x) dx . sec" (3y)tan?(3y)dy C. Involving Partial fractions 4 z? + 2z + 3 1) $77 dx 10 S2-6922+4) dz x2 + 5x - 14 4x - 11 dx x3 - 9x2 8+ t + 6t2 - 12t3 (3t2 + 4)(t2 + 7) dt D. Involving Roots • S2+ve dx 2 + x - 4 E. Involving Trigonometric substitution OSV1– 7yż dy DO 1 F. Improper Integrals Determine if the following integral converges or diverges. If the integral converges, determine its value. 6w2 dw ii) | dx (w+ + 1)2 G. Determine if the following series is absolutely convergent, conditionally convergent or divergent. (-1)+1(n + 1) i) ii) vn n3 + 1 os iii) | V6 – ydy. -00 DO DO üz) (-2)*** 7 E3 n=1 NE 2

Answer 1

ci (3t+t²) sinat dt 오 3+ + +?) -C420 |-63+2+) (3+2+) Cos 24 dt T = (3t+t²) (032+ + 2 | (3124) Sinz Sinat Z Jt 2 Il - (3+++?) (usat + ☆ (3+2+) Sin 24 + (os2t +C tc 2 - - C3+++? 1 Cos 2t + 2 to (3+2+) fin2ttc Aus 22 (ii) ſe? (~ Cez) 2 let I = Seal (3 ( *p 2) 02 I Ī dz D 고 I=4e22 sin (12) s 8 sin (Z) e² u sin (22 e²2 - 8 Ene? 8 cos ( 2 ) eZz, z u fin (Eje? + 32 é%cu(z) - 64 Seascals) 8 868C3) - 5- e 22 こり I - 4 sin (Z) e 22 + 32 é? Co (3) -64 I C 65 I H sin ( 2 ) e ²2 + 32 2² ( (34) 4 sin (Z) e 22 e2? + 32 e?(( a1 22 I- + Ay 65

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