MT1710 - 2018/19: WEEK 8 Chapter III: Integration (continuation). Section 3.2. - Techniques of integration (continuation). Sub-section 3.2.3 - Integration by parts (from last week) Given u and v, two differentiable functions of x, the rule for the differentiation of the product of these two functions is, dx d (uv) = ud2 + val. dx dx dv - + va dt , Then, integrating, uv = dt dt du absorbing the constant into the integral. Hence we get the well-known formula for integration by parts dv dt u- dt = [u ]^ _ 150 dt du dt Examples 3.10 Evaluate, for a and n real constants: (i) teat dt , (ii) |t2 eat dt , (iii) |t2 cos nt dt , (iv) (a2-t2) / dt . (v) |ecostat Workout: (i) In the lecture. (ii) Take u = t2, du = eat dt Then du = 2t, v = e, and using integration by parts and the result from (i), teat dt a 2 = x2 eax - Jedt = 28 2 ] ? a ( xear a eax a2 eaz) + C
where C is a constant of integration. (iii) t2 cos nt dt , Take u = t2 and du = cos (nt) , with then du = 2t and v = 1 sin (nt) . Then evaluate the given integral by parts. Note: You can also evaluate this integral by recalling Euler's formula and using, (iv) Take t2 cos (nt) = Re Je teint dt . L ["(a2 -t2)= dt . u = (a2 - (2) 1/2 , dv =1 dt due = - t (a2 - +2) +2, v = t. Then using integration by parts, I = |(a2-t2) 1/2 dt I = x (a2 - x2) 1/2 - (-+2) (a2-(2)-1/2 dt I = x(a2 - 2) 1/2 1 (-t2 + a2 - a2) (a2 - +2) 1/2 dt I= x(a2 - 2)1/2 - I + a2 1 (a2 - +2) 1/2 dt 21 = x(a2 - x2) 1/2 + a2 arcsin ( a ") + C, with C a constant of integration. (v) In the lecture.
Sub-section 3.2.4 - Partial fractions résumé. Take f (x) =; Q ( x ) P (x) where P (x) is of lower degree than Q (x) (otherwise use long division). Various cases can arise Factor of Q (x) Corresponding partial fractions (a)Unrepeated linear factor (x - a) ? A (b)Repeated linear factor (x - a)k ? A1 + A2 + ... + (c) Unrepeated quadratic factor (x2 + ax + b) ? Bx+ C (d)Repeated quadratic factor (x2 + ax + b)k ? B1x + C1 + . · + BRx + Ck (x -a) (x - a) (x -a)2 (x -a)k Ak (x2 + ax + b) (x2 + ax + b) (x2 + ax + b)k Examples 3.11 Express in partial fractions: 1 x(x-1)(x-2)(x- 3) ' (i) x2 (x - 1)3(x-2) ' 1 (ii) (iii) x4- 2x3 + 2x2 - 2x + 1' (i) and (ii) Workout in