Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUS◮Evaluate the integral I = ∫ cos 4 xdx.Rewriting the integral as a power of cos 2 x and then using the double-angle formulacos 2 x = 1 (1+cos2x) yields2∫∫ ( ) 2 1+cos2xI = (cos 2 x) 2 dx =dx2∫1= (1+2cos2x 4 +cos2 2x) dx.Using the double-angle formula again we may write cos 2 2x = 1 (1+cos4x), and hence2∫ [I = 1 + 1 cos 2x + 1 (1+cos4x)] dx4 2 8= 1 x + 1 sin 2x + 1 x + 1 sin 4x + c4 4 8 32= 3 x + 1 1sin 2x + sin 4x + c. ◭8 4 322.2.5 Logarithmic integrationIntegrals for which the integrand may be written as a fraction in which thenumerator is the derivative of the denominator may be evaluated using∫ f ′ (x)dx =lnf(x)+c. (2.32)f(x)This follows directly from the differentiation of a logarithm as a function of afunction (see subsection 2.1.3).◮Evaluate the integral∫I =6x 2 +2cosxx 3 +sinxdx.We note first that the numerator can be factorised to give 2(3x 2 +cosx), and then thatthe quantity in brackets is the derivative of the denominator. Hence∫ 3x 2 +cosxI =2x 3 +sinx dx =2ln(x3 +sinx)+c. ◭2.2.6 Integration using partial fractionsThe method of partial fractions was discussed at some length in section 1.4, butin essence consists of the manipulation of a fraction (here the integrand) in sucha way that it can be written as the sum of two or more simpler fractions. Againwe illustrate the method by an example.64