Workshop_Solutions_5

1 0 2 5 mathematical methods for physics 1 solutions

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Unformatted text preview: 1 = 2 (1 + x)1/2 x 0 − Note that you should always simplify you result as much as possible. 1 0 2 5 Mathematical Methods for Physics 1, Solutions to Workshop Sheet 5 6. Use the substitution x = sinh(y ) to evaluate the definite integral 1 I= 0 √ 1 x2 + 1 dx. Solution: We use the substitution x = x(y ) = sinh(y ) dx = cosh(y ) = dy ⇔ y = y (x) = arcsinh(x), √ 2 sinh(y ) + 1 = x2 + 1 ⇒ 2 where we have used cosh(y ) − sinh(y ) 2 y (0) = arcsinh(0) = 0 √ 1 x2 + 1 dx = dy, = 1. With this substitution and the new boundaries and y (1) = arcsinh(1), we find that 1 I= 0 √ arcsinh(1) 1 x2 +1 dx = 0 arcsinh(1) 1 dy = y |0 = arcsinh(1). 2 2...
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This document was uploaded on 01/31/2014.

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