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15%20-%20Definite%20Integration

# 15%20-%20Definite%20Integration - PROBLEMS 15 DEFINITE...

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PROBLEMS 15 - DEFINITE INTEGRATION Page 1 Obtain the following integrals as the limit of a sum: ( 1 ) 1 2 0 ( x 3 ) dx + 10 Ans: 3    ( 2 ) 3 x 1 e d x 3 Ans: e - ( 3 ) 3 2 2 2x ) dx - 4 Ans: 3 ( 4 ) 2 x 1 3 d x 6 Ans: log 3 ( 5 ) log 5 x log 2 d x [ Ans: 3 ] ( 6 ) b a cos x dx [ Ans: sin b - sin a ] Solve the following problems: ( 7 ) 3 1 2x 1 dx l l - - 17 Ans: 2 ( 8 ) 2 0 sin mx sin nx dx π , m, n N [ Ans: 0 if m n, π if m = n ] ( 9 ) 1 1 0 x sin dx x 1 + - Ans: 1 2 π - ( 10 ) 1 2 1 3 2 0 2 sin x dx ( 1 x ) - - 1 Ans: log 2 42 π - ( 11 ) 22 2 2 0 x dx a sin x b cos x + π 2 Ans: 2 ab π ( 12 ) 35 0 cos x sin x dx π [ Ans: 0 ] ( 13 ) P. t. 0 2 x tan x dx sec x cos x 4 = + π π ( 14 ) P.t. 0 4 log ( 1 tan x ) dx log 2 8 += π π ( 15 ) P.t. 2 0 2 dx 1 log ( 1 ) sin x cos x 2 =+ + π ( 16 ) P. t. 0 x dx 1 sin x = + π π

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PROBLEMS 15 - DEFINITE INTEGRATION Page 2 Solve the following problems: ( 17 ) If f ( x + α ) = f ( x ) x R, i.e., if f has a period α , then prove that n 00 f ( x ) dx n f ( x ) dx αα = ∫∫ , where n N.
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15%20-%20Definite%20Integration - PROBLEMS 15 DEFINITE...

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