Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUS(c) [(x − a)/(b − x)] 1/2 .2.38 Determine whether the following integrals exist and, where they do, evaluatethem: ∫ ∞∫ ∞x(a) exp(−λx) dx; (b)0−∞ (x 2 + a 2 ) dx;∫ 2 ∞∫11(c)1 x +1 dx; (d) 10 x dx;∫ 2 π/2∫ 1x(e) cot θdθ; (f)dx.00 (1 − x 2 )1/22.39 Use integration∫by parts to evaluate the following:y∫ y(a) x 2 sin xdx; (b) x ln xdx;(c)0∫ y0sin −1 xdx;(d)1∫ y1ln(a 2 + x 2 )/x 2 dx.2.40 Show, using the following methods, that the indefinite integral of x 3 /(x +1) 1/2 isJ = 235 (5x3 − 6x 2 +8x − 16)(x +1) 1/2 + c.(a) Repeated integration by parts.(b) Setting x +1=u 2 and determining dJ/du as (dJ/dx)(dx/du).2.41 The gamma function Γ(n) is defined for all n>−1 byΓ(n +1)=∫ ∞0x n e −x dx.Find a recurrence relation connecting Γ(n +1)andΓ(n).(a) Deduce (i) the value of Γ(n +1) when n is a non-negative integer, and (ii)the value of Γ ( ) (72 ,giventhatΓ 1) √2 = π.(b) Now, ( ) taking factorial m for any m to be defined by m! =Γ(m + 1), evaluate−32 !.2.42 Define J(m, n), for non-negative integers m and n, by the integralJ(m, n) =∫ π/20cos m θ sin n θdθ.(a) Evaluate J(0, 0), J(0, 1), J(1, 0), J(1, 1), J(m, 1), J(1,n).(b) Using integration by parts, prove that, for m and n both > 1,J(m, n) = m − 1n − 1J(m − 2,n) and J(m, n) = J(m, n − 2).m + n m + n(c) Evaluate (i) J(5, 3), (ii) J(6, 5) and (iii) J(4, 8).2.43 By integrating by parts twice, prove that I n as defined in the first equality belowfor positive integers n has the value given in the second equality:∫ π/2I n = sin nθ cos θdθ= n − sin(nπ/2) .0n 2 − 12.44 Evaluate the following definite integrals:∫ ∞[(x 3 +1)/(x 4 +4x +1) ] dx;(a)(c)0 xe−x dx; (b) ∫ 10∫ π/2[a +(a − 1) cos θ] −1 dθ with a> 1 ; (d) ∫ ∞0 2 −∞ (x2 +6x +18) −1 dx.80