MTL 100 Calculus Tutorial Sheet 3 PDF

MTL 100 (Calculus) Tutorial Sheet No. 3 Improper Integral and Beta-Gamma Functions 1. Discuss the convergence of the following improper(proper)integrals. If convergent find their exact va...

MTL 100 (Calculus) Tutorial Sheet No. 3 Improper Integral and Beta-Gamma Functions 1. Discuss the convergence of the following improper(proper)integrals. If convergent find their exact values or an upper bound on the value of the integrals: R ∞ dx R ∞ dx R∞ dx (a) 0 1+x2 ; (b) 1 ; xa (c) −∞ 1+x2 R∞ dx R ∞ x+1 R ∞ sin x (d) 1 x2 (1+ex ) ; (e) 1x3 √ dx; (f) 1 x3 R1 R 1 dx R1 (g) 0 √ dx ; (h) −1 x2 ; (i) 0 √ dx 2 x+4x (1−x) R1 R1 R∞ (j) 0 x a−1 −x e dx; (k) 0 √ xdx ; (l) 0 dx a2 +x2 (1−x2 ) R1 R∞ R∞ (m) log x dx; (n) x sin x dx; (o) √dx 0 0 1 x x2 −1 R ∞ −ax (p) e0 sin bx dx (a > 0). 2. Use Beta and Gamma functions to evaluate evaluate the following: Ra 3 3 R ∞ −x2 (a) x (a 0 − x3 )5 dx; (b) e0 dx π √ (c) 1 xm (log( 1 ))n R R 0 x dx; (d) 0 2 tan x dx π R1 √1 ; √ 1 R (e) 0 2 cos x (f) 0 1−x3 dx π π sin4 θ cos6 θ dθ; cos2x−1 θ sin2y−1 θ dθ. R R (g) 0 2 (h) 2 0 3. Show that (i) β(p,q+1) q = β(p+1,q) p = β(p,q) p−q (ii)β(p, q) = β(p + 1, q) + β(p, q + 1) (iii)β(m, n)β(m + 21 , m + 21 ) = πm−1 21−4m (iv)β(x, x) = 21−2x β(x, 12 ) √ (v)Γ( 41 )Γ( 34 ) = 2π π (vi)Γ(p)Γ(1 − p) = sin πp. 4. Show that R1 (i) 0 sinn θ cosn θ dθ = 12 β( p+1 2 , q+1 2 ) R 1 m−1 (ii) x0 (1 − x2 )n−1 dx = 12 β( m2 , n) Rp m q p qn+m+1 (iii) x (p 0 − xq )n dx = q β(n + 1, m+1 q ) R∞ xp−1 (iv) 0 (1+x)p+q dx = β(p, q) R∞ xm−1 1 (v) 0 (a+bx)m+n dx = an bm β(m, n) R 1 xm−1 +xn−1 (vi) 0 dx (1+x)m+n = β(m, n) m−1 n−1 (vii) ∞ x +x dx R 0 (1+x)m+n = 2β(m, n)  R 1 xl−1 (1−x)m−1 β(l,m) (viii) 0 (b+cx)l+m dx = (b+c)l bm π R cos2m−1 θ sin2n−1 θ β(m,n) (ix) 2 0 (a cos2 θ+b sin2 θ)m+n dθ = 2am bn. R1 p 5. Express In = x (1 − xq )n 0 dx, where p, q, n are positive in terms of Gamma functions. 6. Evaluate R 1 (1−x4 )3/4 (i) 0 (1+x4 )2 dx R2 4 (ii) x (8 − x3 )−1/3 dx 0 R1 (iii) √xdx. 0 1−x5 7. Show that Rπ sinn−1 x 2n−1 (i) 0 (a+b cos x)n dx = (a2 −b2 )n/2 β( n2 , n2 ), Rπ n!nx (ii) 0 (1 − nt )n tn−1 dt = nx β(x, n + 1) = x(x+1)···(x+n) √ 1 2−n [Γ( 1 )]2 Γ( n ) √ dx n = π R1 (iii)0 1−x n Γ( 12 )+ n1 ) = 2 n nΓ(n 2 ) n R1q 1 q 2 1 2 4 (iv) 0 (1 − x )dx = 12 ( π )[τ ( 4 )] R 1 −1/3 (v) x 0 (1 − x)−2/3 (1 + 2x)−1 dx = ( 2π 27 )31/6. 8. Prove that π π √ √dθ × 2 R R (i) 3 0 sin θ 0 sin θ dθ = π R ∞ √ −y2 R ∞ e−y2 π (ii) 0 ye dy × 0√ y dy= 2√ 2 R∞ 2 R ∞ 2 −x4 (iii) 0 xe−x dx × 0 xe dx= 16π√2 π π sinp xdx sinp+1 xdx = π R R (iv) 2 0 2 0 2(p+1). 9. Show that the perimeter of the lemniscate r2 = 2a2 cos 2aθ is √a [Γ( 1 )]2. π 4 10. If n is a positive integer, prove that the ratio of the areas enclosed by the curves x2n +y 2 = 1, 2 1/n x2n + y 2n = 1 is n (n+1). Rπ dx 11. Find the value of 0 a+b cos x a > 0, |x| < a and deduce Rπ dx πa Rπ cos xdx −πb 0 (a+b cos x)2 = (a2 −b2 )3/2 and 0 (a+b cos x)2 = (a2 −b2 )3/2. R ∞ −ax R1 n 12. Starting from e dx 0 = a1 , a> 0and 0x dx = 1 n+1 n > −1, R ∞ m −ax m! R1 n (−1)m m! deduce that 0 x e dx = am+1 and 0 x (log x)m dx = (n+1)m+1. Rx 13. Starting from a suitable integral, show that dx 0 (x2 +a2 ) = 1 2a tan−1 x a + x 2a2 (x2 +a2 ). 14. Show that R ∞ −ax2 q 1 π (i) e0 dx = 2 a R ∞ dx π (ii) 0 x2 +a = √ 2 a ; (a > 0) 2  R ∞ 2n −ax2 √ π 1.2···(2n−1) (iii) x e 0 dx = 2. 2n an+1/2 R∞ √ dx π π 1.2···(2n−1) (iv) 0 (x2 +a)n+1 =. 2 2 2n n!an+1/2 R ∞ −xy sin x (v) 0 e dx x = π 2 − tan−1 y, y > 0 15. Prove that Rπ q (i) 0 log(1 + a cos x)dx = π log[ 12 + 1 2 (1 − a2 )], |a| ≤ 1 R π log(1+a cos x) (ii) 0 cos x dx = π sin−1 a, |a| < 1 π R log(1+cos α cos x) π 2 −4α2 (iii) 0 2 cos x dx = 8. π sin θ cos−1 (cos αcosecθ)dθ = π2 (1 − cos α). R (iv) Π 2 −α 2 π q log(1 − x2 cos2 θ)dθ = π log 1 + (1 − x2 ) − π log 2, if x2 ≤ 1 R (v) 0 2 R ∞ −ax m−1 (vi) e x 0 cos bx dx= Γ(m)rcos m mθ ,where r2 = a2 + b2 and θ = tan−1 ( ab ) R ∞ cos x q R ∞ sin x π (vii) 0 √ dx x = 2 = 0 √ dx. x π log( a+b sin θ R 16. Evaluate 0 2 a−b sin θ ) cosecθ dθ(a > b). R ∞ −ax sin αx 17. If I(α) = e0 dx x then show that I 0 (α) = 1 1+α2. Hence prove that I(α) = tan−1 (α). π log(1+y sin2 x) by showing that I 0 (y) = √ π R 18. EvaluateI(y)= 0 2 sin2 x dx. 2 (1+y) Rx 19. Show that y= k1 0 f (t) sin k(x − t)dt,satisfies the differential equation, d2 y dx2 + k 2 y = f (x),where k is a constant. R ∞ −x cos θ n−1 R ∞ −x cos θ n−1 20. Let v= 0 e x sin(x sin θ)dx, u = 0 e x cos(x sin θ)dx du d2 u 2 Prove that dθ = −nv, dv dθ = −nu and dθ2 + n2 u = 0, ddθv2 + n2 v = 0 Deduce that u=Γ(n) cos nθ, v = Γ(n) sin nθ. -END- 3