The Chemistry Maths Book, Second Edition

170 Chapter 6Methods of integration EXAMPLES 6.7Integrals of type 3 :. (i) , where ais an arbitrary number (buta 1 ≠ 1 − 1 ). Le ...

6.3 The method of substitution 171 Integral 1 in the table is evaluated by means of the substitutionx 1 = 1 a 1 sin 1 θ. Then dx ...

172 Chapter 6Methods of integration Definite integrals When the variable of integration of a definite integral is changed, fromx ...

6.4 Integration by parts 173 6.4 Integration by parts In the integral the integrand is the product of two quite different types ...

174 Chapter 6Methods of integration and (6.14) becomes = 1 x 1 sin 1 x 1 + 1 cos 1 x 1 + 1 C The art of integration by parts is ...

6.4 Integration by parts 175 In general, a polynomial of degree ncan be removed by nsuccessive integrations by parts. The except ...

176 Chapter 6Methods of integration Then, solving for I, and 0 Exercises 49–51 6.5 Reduction formulas The method of integration ...

6.5 Reduction formulas 177 More important still, the recurrence relation is ideally suited for the computation of one or several ...

178 Chapter 6Methods of integration Solving forI n then gives 0 Exercises 52–54 EXAMPLE 6.15Determine a reduction formula for th ...

6.6 Rational integrands. The method of partial fractions 179 Whenn 1 = 11 , Whenn 1 = 10 , (6.18) This last integral is a standa ...

180 Chapter 6Methods of integration EXAMPLE 6.17Integrate. The integrand can be expressed in terms of partial fractions as There ...

6.6 Rational integrands. The method of partial fractions 181 Ifu 1 = 1 f(x) 1 = 1 x 2 1 + 1 px 1 + 1 qthendu 1 = 1 f′(x) 1 dx 1 ...

182 Chapter 6Methods of integration EXAMPLE 6.19Integrate. The quadratic functionx 2 1 + 12 x 1 + 15 can be written as(x 1 + 1 1 ...

6.6 Rational integrands. The method of partial fractions 183 The general form: The numerator can be written as (6.28) so that th ...

184 Chapter 6Methods of integration gives (6.33) wheref(θ)in terms of tis a rational function of t. 0 Exercise 72 EXAMPLE 6.21Ex ...

6.7 Parametric differentiation of integrals 185 In addition, by integration by parts, It follows that so that the order of integ ...

186 Chapter 6Methods of integration EXAMPLE 6.23Integrate. The integral cannot be evaluated by the standard methods discussed in ...

6.8 Exercises 187 6.8 Exercises Section 6.2 Evaluate the indefinite integrals: Evaluate the definite integ ...

188 Chapter 6Methods of integration (u 1 = 1 sin 1 x) 28. (x 1 = 1 sin 1 θ) 31 (i)Use the substitutionx 1 = 1 a 1 sinh 1 ...

Section 6.5 52.Determine a reduction formula for where nis a positive integer. 53.Show that, for integersm 1 ≥ 10 andn 1 ≥ 11 , ...