MT171: Question Sheet 7 - 2018/19: Hints to Solutions 1. Either so that, taking logarithms, Differentiating this, Multiplying through by y gives Or: y = exp[arcsin x] , ln y = arcsin x . y dx 1 dy 1 - x2 1 . dy dx v1-x2 y . dx dy V1-x2 exp[arcsin x] = y V1 - x2 . Either, multiplying through by the square root and squaring: Differentiating this, dy (1 - 2-2) (c)2 = y2 . (1 - x2)2- dx dx2 2x( dx 2 2y Jr . dy ?2y dy = dy Cancelling through by 2- dy # 0 gives the required answer. Or simply multiplying through by the square root and differentiating: -2(1 - x2)- ¿ dy + (1 - x2) 9 ?2y dr2 dy dx = (1 - x2)¿ dy dx i.e. (1 - 22) day - x dx2 dy dx =y. Differentiating twice and substituting into the equation is possible but unattractive! 2. so that, taking logarithms, Differentiating this, 1+x y = \ 1 - x , Iny=>[ln(1+x) - ln(1- x)] . + 1 21+2 1 1-x 1 -1 y dx 1 dy = 1 1-x2 ' so that (1 - x2) dy dx = y,
or working directly. Hence, using Leibniz and differentiating (n - 1)- times to give the nth derivative as the highest derivative: (1 - x2) dny dxn dn-1y +(n-1)(-2.2) den-1+ 2(n-1)(n-2)(-2) 1 dn-2y dn-1y dan-2 drn-1 so that rearranging (1 - x2) Irm - [2(n - 1)x + 1] d"-ly - (n- 1)(n-2) d"-2y =0. dan dxn-1 dan-2 3.(a) .3 x - 1 dx = 0 (1 -x) dx +|(x - 1) dx 1 .3 2 2 1 = x 2 -70 + 1 r2 - x 3 = 2 +2 =3 5 2 . (b) Letting f(x)=(2-x)x(2+x) f(-x)=(2+x)(-x)(2-x) =- f(x) so that, as the interval of integration is symmetric, ·2 (2-x)x(2+x) dx = 0. -2 (c) Jo f(x) da = (23 - 1) dar + .2 x dx = ,=] + [,1 ? 3 = 112 (d) Letting f(x) = x COS x 1+ x4 =- f(x) -x cos x 1+ x4 f(-x) = so that, as the interval of integration is symmetric, 2 x COS x dx= 0. ? 12 1+ x4 (e) 3 x2 - x dx x(x-1) dx + 3 x-1 0 x-1| 1 xx- 1) dx 0 1-x 3 Jo 2 x 1 = ? 1 2 = = ? 1, 1 x dx 72 3 + = ? 2 2 1 7 2 .
4(a) Using e3t = u so that 3e3t dt = 1, e3t 1 + e6t 1 ;dt 3 1 1+ u2 du 3x 1 = arctan u]e 3 = 1 3 arctan e3ª + c , for arbitrary constant of integration c. (b) Using u = cost so that sint dt = - 1, sin t COS x dt 3+2u du 3 + 2 cost = 1 ? -[In |3 + 2u]] cos x letting v = 3 + 2u