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 fo
∫
(lnx)3 dx.
Check your answer by differentiation.
(c) Evaluate
∫ e2
e
(lnx)3 dx.
[ XXXXXXXXXX = 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
[ XXXXXXXXXX = 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]

mme2pse12021-3ubucmcp.pdf

Answered Same Day Oct 17, 2021 University Of South Australia

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Aparna answered on Oct 17 2021

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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...

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