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Differentiation Under the Integral Sign 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.
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