DHS Integration & Its Applications (9758) Topical Revision

7. Integration and its Applications 1 7. Integration and its Applications 1 CJC/2010Promo/10 Find (a) d3 x xx [2] (b) 2 ln dx x x [3] (c) 2 3 d4 7 x xx x    [4] (d) 2 2 1 d 4 x x x    by using the substitution x = 2 sin . [4] 2 DHS/2009Promo/9 (i) Find d (sin 2 ).d xx [1] (ii) Find   2 sin cos d . cos sin x x x x x   [2] (iii) Hence find the exact value of   6 2 0 sin sin 2 cos sin 2 d cos sin x x x x x x x    . [4] 3 ACJC/2010Promo/9 Find the following integrals. (a) 2cos3 cos 3 sin 3 cot 3 x ec x dxx x   [2] (b) 2 1 1 16 x dx x    [4] (c)   2 1 lnx x dx   [4] 4 TJC/2009Promo/2 Integrate the following: (a) 1 x ln x2 dx , where x > 0. [2] (b) 2 1e d 2 1 x x x   , where x > 1 2 . [2]

7. Integration and its Applications 2 5 Using the property    4 2 2sec sec ,sec   find 4sec d  . [3] 6 JJC/2010Promo/12 (a) Write down constants A and B such that, for all values of x , 4 (2 6)x A x B    . [2] Hence find 2 4 d 6 13 x x x x    . [4] (b) Using the substitution 1x u , show that 1 1 4 2 3 1 2 4 1 e d d ex ux u x u  . [2] Hence evaluate 1 4 3 2 1 e dx x x , giving your answer in exact form. [3] (c) Find the exact value of 0 3 3 2 dx x   . [3] 7 RVHS/2010Promo/11 (modified) (a) Using the substitution exu , find 1 de 2ex x x . [4] (b) Show that 1 20 4 5 3 π d = 63 2 x a b x x x      , where a and b are constants to be found. [6] 8 VJC/2013Promo/5 (a) (i) Prove that   22 2 2 d 2 1 d 1 1 1 x x x x x          . [2] (ii) Find the exact value of   1 22 0 1 d . 1 x x  [3] (b) Find the constant A such that Hence find [3] 2 2 2 1 e .1 e 1 e x x x A   2 1 d .1 e x x

7. Integration and its Applications 3 9 HCI/2020Prelim/I/6 Find (a) 3e d5 0.3e x x x , [2] (b)  cos ln dx x , where 0x , [3] (c) the exact value of 3 0 2e 5 dx x . [3] 10 NJC/2020Promo/6 (i) Find 2 2 d 1 x x k x   , where k is a positive constant. [2] (ii) Hence, find   1 2 2 sin d 1 xkx x k x    . [3] (iii) Evaluate   1 0 2 1 2 sin d 1 xx x x     , giving your answer in the form 1 π1 ba       , where a and b are integers to be determined. [2] (iv) Deduce the exact value of     2 1 2 1 sin d 1 m m x mx m x x m            for any constant m. Explain your answer. [2] 11 TJC/2014Promo/6 (a) Using an algebraic method, find the exact value of 4 1 2 dx xx   . [3] (b) Sketch and shade the finite region bounded by the curve y = x2 + 2, the lines y = x and x = 1, and the y-axis. Find the exact volume of the solid formed when the region is rotated 2 radians about the y-axis. [4]

7. Integration and its Applications 4 12 ACJC/2012Promo/15 The shaded region R in the diagram below is bounded by the curve 2 24( 1) ( 2) 4x y    and the lines y x and 1x . (i) Using the substitution cos 1x   , show that 1 2 2 2 d 4x x x      . [5] (ii) Hence, or otherwise, find the exact area of R. [3] (iii) By translating both the curve and the line y x one unit in the positive x-direction, or otherwise, find the volume of the solid formed when R is rotated through 2 radians about the line 1x . [3] 13 NJC/2009Promo/10 (a) (i) Find the derivative of 2 2ex x . [1] (ii) Hence, find   23 21 e dx xx x  . [3] (b) The diagram above shows part of the graph of a curve C given by 1 yx y  . The region R is bounded by C and the lines y x and 4x . Write down the equation of the curve obtained when C is translated by 4 units in the negative x-direction. Hence, or otherwise, write down also an expression for the volume V of the solid formed when R is rotated 2π radians about the line 4x and find its numerical value, giving your answer correct to 3 decimal places. [5] R 4 x y x O 1 yx y  y

7. Integration and its Applications 5 14 RI/2009Prelim/I/9 The diagram above shows the graphs of 23 4y x  and 2y x . Find, in exact form, the coordinates of the points of intersection A and B. [2] The shaded region R is bounded by the two curves and the x-axis. (i) Find the area of R, giving your answers correct to two decimal places. [3] (ii) Find the exact volume of the solid of revolution formed when R is rotated through π radians about the y-axis. [3] 15 DHS/2010Promo/11 (a) The shaded region R in the diagram below is bounded by the curves sin 2y x and cos .y x Find the x-coordinate of the point of intersection P of the two curves. Hence, by integration, find the exact area of R. [5] P y x O π 2 sin 2y x cosy x R 1 23 4y x  2y x A B y x O

7. Integration and its Applications 6 (b) The diagram below (not drawn to scale) shows two regions S and T. The region S is bounded by part of the curve 2 ,4 xy 2x and the x-axis. The region T is bounded by part of the curve 2 4 ,y x 2,x x b and the x-axis. Find the value of b if the area of region S is equal to the area of region T. [4] Let SV and TV be the volume of the solid of revolution formed when the region S and region T are rotated through 4 right angles about the x-axis respectively. Let SW be the volume of the solid of revolution formed when region S is rotated through 4 right angles about the y-axis. Find the value of b if 1 .2 S T SV V W  [5] 16 JJC/2010Promo/10 Region R is bounded by the curves 3y x , 2 4 5 xy  , 3y  , and the axes as shown in the diagram above. (i) Verify that the curves 3y x and 2 4 5 xy  intersect at (3,1). [1] (ii) Find the area of the region R . [3] (iii) Find the volume of the solid generated when R is rotated 2 radians about the xaxis. [3] (iv) Find the volume of the so