7B_2016-2017_H2_Maths_App_of_Integration_Assignment_1_Solutions

National Junior College Mathematics Department 2016 Applications of Integration (Definite Integrals) Page 1 of 3 National Junior College 2016 – 2017 H2 Mathematics Applications of Integration (Definite Integrals) Assignment 1 Solutions 1(i) [B1] Shape (with curve passing through origin). [B1] Equation of asymptote y = 0. 1(ii) Observe from (i) that as ,x 2e 0.xx   Using integration by parts, 2 2 2 1 11 22 1 22 2 1e d e e d 22 110 e e 24 11ee24 3 e.4 x x x x xx x x                             Therefore 30, . 4ab [B1] 2e 0.xx   [M1] Attempt to use integration by parts. [A1] Correct choice of part of integrand to integrate and differentiate. [A2,1,0] 30, . 4ab 2 3 23 2 2 21 2 23 3 11 2 1 2 1 1 11 dd4 1 1 tan sin22 1 tan6 2 2 3 6 tan 0 2 0. a a xxx x x x a a a                      [B1] 11 tan .22 x [B1] 1sin . x [M1] Substitution. [A1] 0.a

National Junior College Mathematics Department 2016 Applications of Integration (Definite Integrals) Page 2 of 3 3 22 dsec 2sec tan .d xx      π 4 23 2 42π2 4 π 3 2π 4 π 23 π 4 π 3 π 4 π 3 π 4 1 2sec tandd 1 sec sec 1 2 dsec 2cos d cos 2 1 d sin 2 2 3 π 1 π 4 3 2 4 π 3 2 .12 4 x xx                              [M1] Attempt to substitute. [A1] π 3 2π 4 2 dsec   [DM1] Integrate 2cos . [A1] π 3 2 .12 4  4(i) Therefore 0.x Alternatively, e e 0xx         2 e 1 0 e 1 e 1 0 e 1 0 e 1 is always +ve e1 ln 0. x xx xx x xx                [M1] Graphical method or express LHS as quadratic:   2 e 1.x  [A1] 0.x

National Junior College Mathematics Department 2016 Applications of Integration (Definite Integrals) Page 3 of 3 4(ii)         3 4 03 40 03 40 4 4 3 3 e e d e e d e e d e e e e 2 e e e e 2 70.75 (to 2 d.p.). xx x x x x x x x x x xx                                   [M1] Split integrand. [A1] [A1]