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09%20-%20Integral%20Calculus

09%20-%20Integral%20Calculus - 9 INTEGRAL CALCULUS Page 1...

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9 - INTEGRAL CALCULUS Page 1 ( Answers at the end of all questions ) ( 1 ) If I 1 = 1 0 2 x dx 2 , I 2 = 1 0 3 x dx 2 , I 3 = 2 1 2 x dx 2 , I 4 = 2 1 3 x dx 2 , then ( a ) I 2 > I 1 ( b ) I 1 > I 2 ( c ) I 3 = I 4 ( d ) I 3 > I 4 [ AIEEE 2005 ] ( 2 ) The area enclosed between the curve y = log e ( x + e ) and the coordinate axes is ( a ) 1 ( b ) 2 ( c ) 3 ( d ) 4 [ AIEEE 2005 ] ( 3 ) The parabolas y 2 = 4x and x 2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S 1 , S 2 , S 3 are respectively the area of these parts numbered from top to bottom, then S 1 : S 2 : S 3 is ( 4 ) dx ) x log ( 1 ) 1 - x log ( 2 2 + is equal to x log 2 + + x 2 + + xe 2 x + + x 2 + + [ AIEEE 2005 ] ( 5 ) Let f ( x ) be a non-negative continuous function such that the area bounded by the curve y = f ( x ), X-axis and the ordinates x = 4 π and x = β > 4 π is ( β sin β + 4 π cos β + 2 β ). Then f ( 2 π ) is π + 2 - 1 π - 2 + 1 π - 2 π + 2 [ AIEEE 2005 ] ( 6 ) The value of dx a 1 x cos x 2 π π + - , a > 0 is ( a ) a π ( b ) 2 π ( c ) a π ( d ) 2 π [ AIEEE 2005 ]
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9 - INTEGRAL CALCULUS Page 2 ( Answers at the end of all questions ) ( 7 ) = n 1 r n r n e lim is ( 8 ) If dx x ( sin x sin ) α - = Ax + B log sin ( x - α ) + C, then the value of ( A, B ) is ( a ) ( sin α , cos α ) ( b ) ( cos α , sin α ) ( c ) ( - sin α , cos α ) ( d ) ( - cos α , sin α ) [ AIEEE 2004 ] ( 9 ) x sin x cos dx - is equal to x tan log 2 1 + π - x cot log 2 1 + 3 2 x tan log 2 1 + π - 3 2 x tan log 2 1 + π + [ AIEEE 2004 ] ( 10 ) The value of dx x 1 3 2 2 - - is 28 14 7 1 [ AIEEE 2004 ]
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