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 sin xx−∫ . [2008 Balestier Hill Sec AMaths P2] 15(a) Express 2)1( )1( + − x xx in the form 2)1(1 ++++ x C x BA , where A, B and C are constants. (b) Evaluate 1 4 0 4 d3(1 2 ) xx−∫ . [2008 Balestier Hill Sec AMaths P2] 16 A curve is such that ( ) 3 12 8 d d − = xx y . The tangent to the curve, at a certain point, cuts the curve at the point (1, 2). Find the equation of the curve. [2008 Catholic High AMaths P1] 17 Differentiate 2 cosxx with respect to x . Hence, or otherwise, evaluate 3 0 sin dx xx π ∫ . [2008 Catholic High AMaths P2] 18 Express )6)(1( 72 2 2 −−+ − xxx x in partial fractions. Hence, or otherwise, find 2 2 27 d.( 1)( 6) x xx xx − + −−∫ [2008 Catholic High AMaths P2] 19 Evaluate π 22 π 4 (sin 2sin 2 ) dx xx−∫ . [2008 Cedar Girls’ Sec AMaths P1]

4 20(a) Express 2)1)(32( 49 −+ − xx x in partial fractions. Hence evaluate 3 22 94 d(2 3)( 1) x xxx − +−∫ . (b) The gradient function of a curve y = f( x) is given by 2d d x nmx y += . The curve passes through the point A (2, 5). The tangent to the curve at point A is parallel to the line 34 =− yx and the curve has a turning point at 2 3x=− . Find (i) the value of m and of n, (ii) the equation of the curve. [2008 Cedar Girls’ Sec AMaths P2] 21(a) Find 2 1 4 1 dcos xx∫ . (b) Express ( )d ln cot 3d xx in the form sin k px , where k and p are constants. [2008 CHIJ St Nicholas Girls’ AMaths P1] 22(a) Express xx xx 4 8183 3 2 − −− in partial fractions. (b) Evaluate 9 4 353 d5 x xxx + +∫ , giving your answer correct to 2 decimal places. [2008 CHIJ Toa Payoh Sec AMaths P2] 23 Differentiate x2 ln x with respect to x. Hence, evaluate 3 2 ln dx xx∫ , giving your answer correct to two decimal places. [2008 CHIJ Toa Payoh Sec AMaths P1] 24(a) Express ( )( ) 2 2 2 16 13 2 1 xx xx + −+ in partial fractions. Hence, find ( )( ) 2 2 2 16 d 13 2 1 xx x xx + −+∫ . (b) Show that ( )d3 121d 21 xxxx x + −= − . Hence, evaluate 13 5 2 d 32 1 x x x−∫ . [2008 CHIJ St Nicholas Girls’ AMaths P2]

5 25(a) Find (i) 5 24 d(3 2) xx−∫ , (ii) 23 (2 1) dxx x −∫ . (b) (i) Given 1)23(5 +−= xxy , show that d 5(9 4) d 21 yx x x += + . (ii) Hence, find 94 d 1 x x x + +∫ . [2008 Clementi Woods Sec AMaths P1] 26(i) Express )1( 1 2 −xx in partial fractions. (ii) Hence, find 2 1 d( 1) xxx −∫ , simplifying your answer. [2008 Clementi Woods Sec AMaths P2] 27(i) Show that d [ln(sec tan )]d xxx + = sec x. (ii) Express d (sec tan )d xxx in terms of sec x. (iii) Hence, evaluate π 33 0 sec dxx∫ , giving your answer in surd form. [2008 Clementi Woods Sec AMaths P2] 28 Given that ( ) , sin1 2 2 θ+ =y find d d y θ . Hence, or otherwise, find the value of k given that ( ) 30 cos 5 d 181 sin k θ θ θ = +∫ and k < 2. [2008 Commonwealth Sec AMaths P2 (modified)] 29 Express 23 346 23 + −+= x xxxy in partial fractions and hence find dyx∫ . [2008 Commonwealth Sec AMaths P2 (modified)] 30(i) Evaluate ( ) π 224 0 cos sin dx xx−∫ . (ii) Evaluate 2 0 42d 32 xx  − +∫ . [2008 Crescent Girls’ AMaths P1 (modified)]

6 31(i) Express 44 5 2 +− − xx x in partial fractions. (ii) Hence, evaluate 6 24 5 d44 x xxx − −+∫ . [2008 Crescent Girls’ AMaths P2] 32(i) Express 12 74 2 2 −+ −+ xx xx in partial fractions. (ii) Hence, evaluate xxx xx d12 745 4 2 2 ∫ −+ −+ . [2008 Holy Innocents High AMaths P2] 33 Evaluate π 6 0 (2cos sin 2 ) dx xx+∫ . [2008 Holy Innocents High AMaths P1] 34(a) Given that ( ) ( ) 36 13 fd fd 7xx xx= =∫∫ , find the value of A where ( ) 6 1 2f dA x xx= + ∫ . (b) Differentiate 2 21 x x− with respect to x, and hence evaluate ( ) ( ) 2 21 1 d 21 xx x x − −∫ . [2008 Maris Stella High AMaths P2 (modified)] 35 Evaluate (i) 1 0 1 d 13 x x+∫ , (ii) ( ) π 3 0 cos3 sin dx xx−∫ . [2008 Ngee Ann Sec AMaths P1] 36 Show that ( ) 2d 2 sin 1 cos 2 2 sin 2d x x xx xx = −+ . Hence, evaluate π π 2 sin 2 dx xx∫ , leaving your answer in terms of π. [2008 Ngee Ann Sec AMaths P2] 37(i) Express 2 2 2 74 32 xx xx ++ ++ in partial fractions. (ii) Hence, evaluate 22 21 2 74 d2 64 xx xxx ++ ++∫ . [2008 Singapore Chinese Girls’ AMaths P2]

7 38 Given that 5 4e xy= − , show that d 2e 0d xyy x+= . Hence, or otherwise, evaluate 1 2 0 ln e d 5 4e x x x −∫ . [2008 Singapore Chinese Girls’ AMaths P2] 39 Given that (sin cos ) ex xxy −= , show that d 2cos de x yx x = . Hence, show that π π0 cos 1 1d1e 2ex x x = +∫ . [2008 Temasek Sec AMaths P2] 40 Evaluate ( ) π 31 2 0 e sec 2 1 dx xx− ++∫ , correct your answer to 2 decimal places. [2008 Temasek Sec AMaths P1] 41 Differentiate ( ) ( )5 ln 5xx++ . Hence, evaluate 3 0 ln( 5) d5 x x+ ∫ , correct to three significant figures. [2008 Unity Sec AMaths P1] 42 Show that ( ) 2 2 2 d