DHS Integration & Its Applications (9758) (Revision Solutions)

Y5 Topical Revision Package 1 7. Integration and its Applications (solutions) 1 7. Integration and its Applications (solutions) 1 (a) (b) (c) (d) CJC PROMO 2010/QN10 ∫ x 3 + x dx = ∫ 3 + x – 3 3 + x dx = ∫ 1 – 3 3 + x dx = x – 3 ln | 3 + x | + c ∫x2 ln x dx = [ x3 3 ln x ] – ∫x3 3 1 x dx = [ x3 3 ln x ] – 1 3 ∫x2 dx = [ x3 3 ln x ] – x3 9 + C ∫ x + 3 x2 + 4x + 7 dx = 1 2 ∫ 2x + 6 x2 + 4x + 7 dx = 1 2 ∫ 2x + 4 + 2 x2 + 4x + 7 dx = 1 2 [∫ 2x + 4 x2 + 4x + 7 dx + ∫ 2 x2 + 4x + 7 dx] = 1 2 [ ln (x2 + 4x + 7) + ∫ 2 (x + 2)2 + 3dx ] = 1 2 [ ln (x2 + 4x + 7) + 2 3 tan–1 x + 2 3 ] + C Let x = 2 sin θ dx = 2 cos θ dθ ∫ 2x – 1 4 – x2 dx = ∫ 4 sin θ – 1 2 cos θ 2 cos θ dθ = ∫ 4 sin θ – 1 dθ = – 4 cos θ – θ + c = – 4 4 – x2 2 – sin–1 x 2 + c = – 2 4 – x2 – sin–1 x 2 + c

Y5 Topical Revision Package 1 7. Integration and its Applications (solutions) 2 2 (i) (ii) (iii) DHS PROMO 2009/QN9 d (sin 2 ) 2cos 2d xxx  ( ) ( ) ( ) ( ) ( ) 2 2 2 1 sin cos d (sin cos ) cos sin d cos sin ( sin cos ) c os sin d cos sin 1 1 cos sin xx x x x x xx xx x x x xx xx C xx − − − + = +− − = −− − − −=−+ − = +− ∫∫ ∫ C ( ) 6 2 0 6 6 00 22 6 0 sin sin2 cos sin2 d cos sin 11sin2 . .2cos2 dcos sin cos sin 1 cos sinsin . 2 d3 cos sincos sin66 xx xx x xx x xxxx xx xx xxx π π π π π ππ + − = − −− −= − −− ∫ ∫ ∫ ( ) [ ] 6 0 6 0 3 2 cos sin d 31 33 2 sin cos2 33 3 22 x xx xx π π = −+ − +=−− = − ∫ 3 (a) (b) ACJC PROMO 2010/QN9 2cos3 cos 3 1 ln sin 3 cot 3sin 3 cot 3 3 x ec x dx x x cxx − = +++∫ ( ) ( ) ( ) 2 22 1 16 1 2 2 2 21 4 11 1 16 4 1 16 11 1 32 1 164 32 xx dx dx dx x xx dx x x dx x − − = − − −− = +− − − ∫ ∫∫ ∫∫ ( ) 12 12 12sin 1 1614 324 11sin 4 1 164 16 x xC x xC − −  = + −+ = +−+

Y5 Topical Revision Package 1 7. Integration and its Applications (solutions) 3 (c) ( ) ( ) ( )∫∫ −−   −=− − dxx x xx dx x x 1 1 1 1lnln1 2 = ∫    +−−− dxx x x x 1 1 1 1 ln = ln ln 1 ln1 x x xCx+ −− +− 4 TJC PROMO 2009/QN2 (a) 2 11 1 11 ln lnln 2 ln 2 ln 2 xxdx dx dx x Cxx x x = = = +∫∫∫ (b) 21 21 211 21 21 x xxe dx e dx e C xx − −−= = + −−∫∫ 5 4 22 31 3(1 tansec sec tan ) tand dCθ θ θθ θ θ+ + += =∫∫ 6 (a) JJC PROMO 2010/QN12 (2 6) 2 6A x b Ax A b+ += + + Comparing coefficients of x , 121 2AA= ⇒= Comparing coefficients of constant term, 34 1BB+=⇒= 14 (2 6) 12xx∴+= + + 2 4 d 6 13 x x xx + ++∫ 2 1(2 6) 12 d 6 13 x x xx ++ = ++∫ 22 1 26 1 d d2 6 13 6 13 x xx xx xx += + ++ ++∫∫ 2 22 1 26 1 d d2 6 13 ( 3) 2 x xx xx x += + ++ ++∫∫ ( ) 211 13ln 6 13 tan2 22 xxx c− += +++ + 