# tut08 - The University of Sydney Math1003 Integral Calculus...

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The University of Sydney Math1003 Integral Calculus and Modelling Semester 2 Exercises for Week 8 2011 Assumed Knowledge Factorisation of expressions. Simple techniques of integration. Objectives (7a) To be able to recognise a di±erential equation as a separable equation. (7b ) To be able to solve a separable equation by separation of variables. (7c) To be able to perform integrations using trigonometric substitutions. Preparatory Questions 1. Which of the following di±erential equations are separable? Write those that are in separated form. ( i ) dy dx = x 2 y ( x 2 + 1) 1 / 2 ( ii ) dy dx = a 2 e y ( a 2 - x 2 ) 3 / 2 - e y ( a 2 - x 2 ) 1 / 2 ( iii ) dy dx = x + cos y x 3 x 2 - 16 . Practice Questions 2. Find the general solutions of ( i ) dy dx = 1 + y 2 ( ii ) dy dx = y cos x ( iii ) (1 + x ) dy dx + y 2 = 0 3. Evaluate the following integrals by making the given substitution: ( i ) i x 2 ( a 2 - x 2 ) 3 / 2 dx, x = a sin u .

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tut08 - The University of Sydney Math1003 Integral Calculus...

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