assignment4bfsol - ma2176 a4 Brief Sol Integration and its...

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1 ma2176 a4 Brief Sol Integration and its Applications 1. Let + < = 2 1 if 2 1 0 if 3 ) ( x x x x x f . Find 2 0 , ) ( ) ( 0 = x dt t f x F x . Solution : For 1 0 < x , () ( ) 2 00 () 3 3 2 xx x Fx f td t td t x == = ∫∫ . For 2 1 x , () () () ( ) 11 2 0010 1 3 ( 2 ) 2 . 2 x x ftd t t t t t x ==+= + + = + ∫∫∫∫ 2. Evaluate the following integrals: (a) dx x x + 1 (b) dx x x + 2 2 1 (c) dx e e x x + + 1 1 3 Solution : (a) C x x dx x x dx x x + + = + = + 2 3 2 1 1 2 3 (b) 2 22 1 1a r c t a n x dx dx x x C  =− = +  ++  (c) ( ) ( )( ) 3 2 3 2 2 111 1 1 1 2 x x x x x x ee e e dx dx dx e e dx e x C e + + = + = + + + 3. Evaluate the following integrals: (a) dx x x 1 2 (b) dx x x + 1 0 10 2 1 (c) 1 0 21 x xd x (d) 4 3 0 sin (2 )cos(2 ) x xdx π (e) dx x x 2 1 (f) dx x tan Solution : (a) 3 2 2 2 1 1 23 x x x x C −= = −+ (b) 10 10 2 1 1 1 2 1 2 d x x dx += + + = (c) 1 0 ( ) 3 d d x x d x ++= + + + = +
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2 (d) () 44 33 00 11 s i n( 2)c o s ( 2) s i 2) s i n ( 28 xx d x x dx ππ == ∫∫ (e) 222 22 2 1 111 2 1 x dx x d x x C x =− =− − + (f) sin (cos ) tan ln cos cos cos xd x xdx dx x C x x = + 4. Evaluate the following integrals by integration by parts: (a) dx xe x (b) xdx x 2 sin 2 (c) xdx arcsin Solution : (a) ( ) x x x ed x x d e x e e C −− = + (b) 2 c o s ( c o s ( c o s ( s i n ( s i n ( c o s ( 2 2 c o s ( s i n ( 2) c o s ( 4 x x x x d x x d x x d x x x C  + = +   + + + (c) 2 2 arcsin arcsin (arcsin ) arcsin arcsin 1 1 x xdx x x xd x x x dx x x x C x =+ + 5. Evaluate the following integrals by partial fractions: (a) 3 2 2 x dx +− (b)
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This note was uploaded on 11/09/2009 for the course ENG 91301 taught by Professor Lui during the Spring '08 term at Hong Kong Institute of Vocational Education.

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assignment4bfsol - ma2176 a4 Brief Sol Integration and its...

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