(a) Evaluate the following integrals (i) (2t + i)² dt. (ii) fot, dt. 2 t+i ㅠ (iii) S 2teit² dt.

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problem 2 (Integration along complex curves)
(a) Evaluate the following integrals
(i) f¹ (2t + i)² dt.
·2
(ii) foi dt.
t+i
(iii) S 2te¹t² dt.
(b) Show that
[ * ƒ(t) f'(t) dt = ¦ ¦ (ƒ² (b) — ƒ²(a)).
(c) Consider the complex function f(x + y) = 2xy
(i) Show that the integral of f over the square whose vertices are 0, 1, 1 + i and i is
non-zero.
(ii) Deduce that f is not the derivative of any function.
Transcribed Image Text:Problem 2 (Integration along complex curves) (a) Evaluate the following integrals (i) f¹ (2t + i)² dt. ·2 (ii) foi dt. t+i (iii) S 2te¹t² dt. (b) Show that [ * ƒ(t) f'(t) dt = ¦ ¦ (ƒ² (b) — ƒ²(a)). (c) Consider the complex function f(x + y) = 2xy (i) Show that the integral of f over the square whose vertices are 0, 1, 1 + i and i is non-zero. (ii) Deduce that f is not the derivative of any function.
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