DHS Y4 Math 2 Supplementary Worksheet_Integration Techniques

1 Year 4 Mathematics 2 Supplementary Worksheet Integration Name : _________________________________ ( ) Class : _______ Date : __________ 1 If 3 1 21 d5x xpx + =∫ , find the value of p . [2008 ACS (Barker Road) AMaths P2] 2 Show t hat ( )d 234d xxx  −− can be written in the form x bxa − − 42 where a and b are integers. Hence evaluate 3 0 57 18 d 4 x x x − −∫ . [2008 ACS (Barker Road) AMaths P2] 3 Evaluate π 3 0 sin (sin 1)x x dx +∫ . [2008 ACS (Barker Road) AMaths P2] 4(i) Express 2 16 2 95 x xx − −− in partial fractions. (ii) Hence evaluate 8 26 16 d2 95 x xxx − −−∫ . [2008 ACS (I) AMaths P1] 5 Evaluate 22 0 1e d e x x x+ ∫ . [2008 ACS (I) AMaths P1] 6 A curve has the equation ln 2yx x= , where x > 0. (i) Find an expression for d d y x . (ii) Hence, find 2 1 3ln 2 dxx∫ . (iii) Find the range of values of x for which the function ln 2yx x= is increasing. Leave your answer in terms of e. (iv) Find the rate of change of x given that y increases at the rate of 4 units/second when x is 1 2 e units. [2008 ACS (I) AMaths P2]

2 7(a) Evaluate (i) 4 3 3 d25 xx−∫ , (ii) 1 0 9 d 23 1 x x+∫ , (iii) π 22 0 4 sin 2 dxx∫ . (b) The gradient of a curve at the point ( x , y) on the curve is given by 23 k x− , where k is a constant. If the tangent to the curve at the point (2 , 8) cuts the y-axis at (0 , 4), find (i) the value of k , (ii) the equation of the curve. [2008 ACS (I) AMaths P2] 8 Show that 22 sin 2 sin 4 tan2 sin 2 sin 4 xx xxx − =+ . Hence evaluate 4 0 2sin2 sin4 d2sin2 sin4 xx xxx π − +∫ , leaving your answer in terms of π . [2008 ACS (I) AMaths P2] 9(a) Express )2( 62 2 2 − −+ xx xx in partial fractions. (b) Hence, find the exact value of 25 23 26 d( 2) xx xxx +− −∫ . [2008 Anderson Sec AMaths P1] 10(a) Express xx cos4cos in the form 2 coscos BA+ and hence find cos4 cos dx xx∫ . (b) Show that 2 d1 d 13 ( 13 ) x xx x  = −− . Hence evaluate 2 6 2 2 d13 xx   −∫ . [2008 Anderson Sec AMaths P2] 11 Show that the gradient function for the curve x xy cos tan= is x x 3 2 cos sin1+ . Hence deduce that the curve has no stationary points for 0 2x π<< . [2008 Anglican High AMaths P1] 12(i) Express )4)(12( 6135 2 2 −+ −+ xx xx in partial fractions. (ii) Hence evaluate 24 23 5 13 6 d(2 1)( 4) xx xxx +− +−∫ . [2008 Anglican High AMaths P2]

3 13 The curve for which 3 d d (2 ) yk xx= − where x > 2 and k is a constant, is such that the tangent at (3 , 6) passes through the origin. Find (i) the value of k , (ii) the equation of the curve. [2008 Balestier Hill Sec AMaths P1] 14 A curve has the equation x xy sin1 cos2 −= . (i) Show that d d y x can be written in the form of x k sin1− where k is a constant. State the value of k. (ii) Hence, evaluate π 6 0 1 d1