MAT137Y1 Midterm: MAT 137Y, 2003-2004 Test 3 Self Generated Solutions.pdf
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MAT137Y1 Full Course Notes
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Mat 137y, 2003-2004 test 3 solutions: compute the following integrals. Z (10%) (i) (10%) (ii) (10%) (iii) (10%) (iv) Let i be the integral and integrate by parts. We let u = (ln x)2, dv = x3 dx. Then du = 2ln x and v = 1. 32 (u = ln x, dv = x3 dx, du = 1 x dx, v = 1 x4 +c. x3 dx (cid:21) Then dx = 4sec2 u du and this gives. 16tan2 u 4sec u dx x2 + 16. Z sec u tan2 u csc u = du = 1. + b (x 1)2 + c x + 1. = a(x 1)(x + 1) + b(x + 1) +c(x 1)2 = 4x. x 1. If x = 1, this gives 2b = 4, or b = 2. If x = 1 we get 4c = 4, or c = 1. A + b +c = 0, or a = 1.
