Created by T. Madas DIFFERENTIATION UNDER THE INTEGRAL SIGN Created by T. Madas Created by T. Madas LEIBNIZ INTEGRAL RULE Created by T. Madas Created by T. Madas Question 1 The function f satisfies the following relationship. x 2 fx= ft dt , f 2 = 1 . ()() () 2 1 Determine the value of f 1 . ( 2 ) SPX-Y , f 1= 2 ( 2 ) 7 Created by T. Madas Created by T. Madas Question 2 Find the value of 3p+ 2 x d x + 6 lim dx . p→0 dp4 x 2p− 1 MM1P , 23 5 Created by T. Madas Created by T. Madas Question 3 Find the general solution of the following equation 2x d 2 sin()()t2+ cos2 tdt 2 = − , x ∈ » . dx1π x 6 SPX-N , x=1 π 4 k − 1 k ∈ » 4 () Created by T. Madas Created by T. Madas Question 4 The function g is defined as b( x ) gx()()= fxtdt, . a() x a) State Leibniz integral theorem for g′( x ) . x d1+ x2 t 2 b) Find a simplified expression for dt . dx t −1 x x 2 2 4 d1+ xt 2 1 + x MM1G , dt = dxx−1 t x Created by T. Madas Created by T. Madas INTEGRATION APPLICATIONS INTRODUCTION Created by T. Madas Created by T. Madas Question 1 It is given that the following integral converges. 1 4 x3 ln xdx . 0 a) Evaluate the above integral by introducing a parameter and carrying out a suitable differentiation under the integral sign. b) Verify the answer obtained in part (a) by evaluating the integral by standard integration by parts. V , MM1A , − 9 49 Created by T. Madas Created by T. Madas Question 2 1 8 dx . 2 1+ x2 0 () Evaluate the above integral by introducing a parameter k and carrying out a suitable differentiation under the integral sign. You may not use standard integration techniques in this question. V , π + 2 Created by T. Madas Created by T. Madas Question 3 4 dx . 2 ()1− 4 x2 Find a simplified expression for the above integral by introducing a parameter a and carrying out a suitable differentiation under the integral sign. You may assume 1 1 x • dx =artanh + constant , x< a . a2− x 2 a a d 1 • ()artanh u = du 1− u2 You may not use standard integration techniques in this question. 2x artanh 2 x+ + C 1− 4 x2 Created by T. Madas Created by T. Madas Question 4 x3e 2 x dx . Find a simplified expression for the above integral by introducing a parameter α and carrying out a suitable differentiation under the integral sign. You may not use integration by parts or a reduction formula in this question. 1 e42x x 3− 6 x 2 + 63 x −+ C 8 Created by T. Madas Created by T. Madas Question 5 1 3 dx . ()5+ 4 x − x 2 2 Find a simplified expression for the above integral by introducing a parameter a and carrying out a suitable differentiation under the integral sign. You may assume 1 x dx =arcsin + constant , x≤ a . a2− x 2 a You may not use standard integration techniques in this question. x − 2 + C 9 5+ 4 x − x 2 Created by T. Madas Created by T. Madas Question 6 It is given that the following integral converges ∞ xne−α x dx , 0 where α is a positive parameter and n is a positive integer. By carrying out a suitable differentiation under the integral sign, show that Γ(n +1) = n ! . You may not use integration by parts or a reduction formula in this question. proof Created by T. Madas Created by T. Madas Question 7 It is given that the following integral converges 1 xm []ln xn dx , 0 where n is a positive integer and m is a positive constant. By carrying out a suitable differentiation under the integral sign, show that 1 n n ()−1n ! xm ln x dx = . [] n+1 0 ()m +1 You may not use standard integration techniques in this question. MM1B , proof Created by T. Madas Created by T. Madas Question 8 π 1 I ()α = dx , α >1. 0 α − cos x a) Use an appropriate method to show that π I ()α = . α 2 −1 b) By carrying out a suitable differentiation under the integral sign, evaluate π 1 dx . 2 2− cos x 0 () You may not use standard integration techniques in this part of the question. π 2 Created by T. Madas Created by T. Madas FURTHER INTEGRATION APPLICATIONS Created by T. Madas Created by T. Madas Question 1 ∞ ln( 1+ 4 x2 ) I = dx . x2 0 By introducing a parameter in the integrand and carrying a suitable differentiation under the integral sign show that I = 2π . V , proof Created by T. Madas Created by T. Madas Question 2 It is given that the following integral converges. 1 x −1 I = dx . 0 ln x Evaluate I by carrying out a suitable differentiation under the integral sign. You may not use standard integration techniques in this question. V , MM1C , ln 2 Created by T. Madas Created by T. Madas Question 3 ∞ e−2x− e − 8 x I = dx . 0 x By introducing a parameter in the integrand and carrying a suitable differentiation under the integral sign show that I = ln 4 . V , proof Created by T. Madas Created by T. Madas Question 4 It is given that ∞ sin ()kx π dx = . 0 kx 2 Use Leibniz’s integral rule to show that ∞ sin 2 x π dx = . x2 2 0 MM1J , proof Created by T. Madas Created by T. Madas Question 5 It is given that the following integral converges. 1 x5 −1 dx . 0 ln x Evaluate the above integral by introducing a parameter and carrying out a suitable differentiation under the integral sign. You may not use standard integration techniques in this question. ln 6 Created by T. Madas Created by T. Madas Question 6 ∞ e−2x sin x I = dx . 0 x By introducing in the integrand a parameter k and carrying a suitable differentiation under the integral sign show that I = arccot 2 . V , MM1E , proof Created by T. Madas Created by T. Madas Question 7 ∞ e−x− e − 7 x I = dx . xsec x 0 By introducing in the integrand a parameter α and carrying a suitable differentiation under the integral sign show that I = ln5 . V , proof Created by T. Madas Created by T. Madas Question 8 ∞ cos x −4x − 6 x I =e − e dx . 0 x By introducing in the integrand a parameter λ and carrying a suitable differentiation under the integral sign show that = 1 ln 2 . I 2 proof Created by T. Madas Created by T. Madas Question 9 ∞ e−x =1 − cos 3 x dx . I ()4 0 x By introducing in the integrand a parameter λ and carrying a suitable differentiation under the integral sign show that I =ln5 − ln 4 . proof Created by T. Madas Created by T. Madas Question 10 It is given that the following integral converges ∞ sin t dt . 0 t Evaluate the above integral by introducing the term e−α t , where α is a positive parameter and carrying out a suitable differentiation under the integral sign. You may not use contour integration techniques in this question. π V , 2 Created by T. Madas Created by T. Madas Question 11 Show, by carrying out a suitable differentiation under the integral sign, that ∞ e−ax sin bx b dx = arctan , x a 0 where a and b are positive constants. You may assume ∞ sin t π dt = . 0 t 2 V , proof Created by T. Madas Created by T. Madas Question 12 Given that a is a positive constant, find an exact simplified value for ∞ ∞ sinxy∂2 sin xy a2 dx− dx . xax22+∂y2 xax 22 + 0() 0 () You may assume ∞ sin t π dt = . 0 t 2 π 2 Created by T. Madas Created by T. Madas Question 13 1 2 n2 x In = xe dx , n = 0, 1, 2, 3, ... 0 By introducing in the integrand a parameter k and carrying a suitable differentiation under the integral sign show that n n e n n ()−1n ! In = () −1r ! − . 2n+1r 2 n + 1 r=0 proof Created by T. Madas Created by T. Madas Question 14 1 2n+ 1 x 2 In = xe dx , n = 0, 1, 2, 3, ... 0 By introducing in the integrand a parameter k and carrying a suitable differentiation under the integral sign show that n en n 1 n In = () −1!r −−() 1! n . 2r 2 r=0 proof Created by T. Madas Created by T. Madas Question 15 ∞ arctan8x− arctan 2 x I = dx . 0 x By introducing a parameter in the integrand and carrying a suitable differentiation under the integral sign show that I = π ln 2 . V , proof Created by T. Madas Created by T. Madas Question 16 It is given that the following integral converges ∞ e−ax− e − bx dx , x 0 where a and b are positive constants. By carrying out a suitable differentiation under the integral sign, show that ∞ e−ax− e − bx b dx = ln . x a 0 V , proof Created by T. Madas Created by T. Madas Question 17 It is given that the following integral converges ∞ cos kx −ax − bx e− e dx , 0 x where k , a and b are constants with a > 0 and b > 0 . By carrying out a suitable differentiation under the integral sign, show that ∞ coskx 1 b2+ k 2 e−ax− e − bx dx = ln . 2 2 0 x 2 a+ k proof Created by T.