MATHEMATICAL METHODS FOR ENGINEERS 2 Problem Solving Exercise 1 1. (a) Use integration by parts to derive the following reduction formula:∫ (lnx)n dx = x(lnx)n − n ∫ (lnx)n−1 dx. (b) Use the reduction...

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MATHEMATICAL METHODS FOR ENGINEERS 2 Problem Solving Exercise 1 1. (a) Use integration by parts to derive the following reduction formula:∫ (lnx)n dx = x(lnx)n − n ∫ (lnx)n−1 dx. (b) Use the reduction formula to find an expression for ∫ (lnx)3 dx. Check your answer by differentiation. (c) Evaluate ∫ e2 e (lnx)3 dx. [3 + 4 + 2 = 9 marks] 2. Solve the following separable ordinary differential equations: (a) (x2 + 2x− 3) dy dx = x + 5 (b) (x2 + 2x + 10) dy dx = 2x3 − x2 − 4x− 1 [3 + 6 = 9 marks] 3. For each of the following improper integrals, determine whether it converges, and if so, evaluate it. (a) ∫ ∞ −∞ sinhx dx (b) ∫ 3 2 dx√ 3− x (c) ∫ ∞ 2 dx x(lnx)2 [3 + 3 + 3 = 9 marks] 4. Solve the following differential equations (a) x dy dx = 2y + x3 cosx. (b) x dy dx = (1 + x)(1 + y), with y (1) = 0. [4 + 4 = 8 marks] [Total 35 marks]