Differentiation Under the Integral Sign

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DIFFERENTIATION UNDER THE INTEGRAL SIGN

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LEIBNIZ INTEGRAL RULE

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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

Question 2 Find the value of

3p+ 2  x  d x + 6    lim   dx  . p→0 dp4 x      2p− 1  

MM1P , 23 5

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Question 3 Find the general solution of the following equation

2x  d 2 sin()()t2+ cos2 tdt 2  = − , x ∈ » . dx1π  x 6 

SPX-N , x=1 π 4 k − 1 k ∈ » 4 ()

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Question 4 The function g is defined as

b( x ) gx()()= fxtdt, . a() x

a) State Leibniz integral theorem for g′( x ) .

x  d1+ 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  = dxx−1 t  x

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INTEGRATION APPLICATIONS INTRODUCTION

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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

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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

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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

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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  

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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

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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

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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

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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

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FURTHER INTEGRATION APPLICATIONS

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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

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

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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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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+1r   2 n + 1 r=0

proof

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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 en  n  1 n In =  () −1!r  −−() 1! n . 2r   2 r=0

proof

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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

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

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

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Question 18 It is given that the following integral converges

1 xa− x b dx ,  ln x 0

where a and b are constants greater than −1.

By carrying out a suitable differentiation under the integral sign, show that

1 xa− x b  a + 1  dx = ln .  lnx b + 1  0

proof

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Question 19 It is given that the following integral converges

∞ sin mx −ax − bx  e− e  dx , 0 x where a , b and m are constants, with m ≠ 0 , a > 0 , b > 0 .

By carrying out a suitable differentiation under the integral sign, show that

∞ sin mx−ax − bx   b  a e− e  dx = arctan − arctan  . 0 x m  m

proof

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Question 20 It is given that the following integral converges

∞ arctan ax dx , a > − 1.  x1+ x 2 0 ()

By carrying out a suitable differentiation under the integral sign, show that

∞ arctan ax π dx=ln() a + 1 .  x1+ x 2 2 0 ()

proof

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Question 21 It is given that the following integral converges

∞ ln( 1 + a2 x 2 ) dx ,  1+ b2 x 2 0 where a and b are constants.

By carrying out a suitable differentiation under the integral sign, show that the exact value of the above integral is

π a+ b ln . b b

MM1F , proof

Created by T. Madas

Question 22 It is given that the following integral converges

∞ e−x− e − 2 x dx . 0 x

a) By introducing a parameter k and carrying out a suitable differentiation under the integral sign, show that

∞ e−x− e − 2 x dx = ln 2 . 0 x

b) Use the result of part (a) and differentiation under the integral sign to show further that

∞ −x e 1 1 −2x  2−+ e  dx =−+ 2ln27 . 0 x x x 

proof

Created by T. Madas

Question 23 The integral function y= y( x ) is defined as

x2 cosx cos θ y x d . () = 2 θ 1 2 1+ sin θ π 16

Evaluate y′(π ) .

2π

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Question 24 An integral I is defined in terms of a parameter α as

∞ α 2  exp x2 dx . I ()α = − − 2   x  0

By carrying out a suitable differentiation on I under the integral sign, show that

∞ 2 1  π exp −x −  dx = .  16 x2  4e 0

proof

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Question 25 An integral I with variable limits is defined as

x2 I ()x= e u du . x

a) Use a suitable substitution followed by integration by parts to find a simplified expression for

d I() x  . dx

b) Verify the answer obtained in part (a) by carrying the differentiation over the integral sign.

d x x MM1D , I() x  =2 x e − e dx

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Question 26 Use complex variables and the Leibniz integral rule to evaluate

1 sin() ln x dx .  ln x 0

You may assume that the integral converges.

MM1H , 1 π 4

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Question 27

∞ 2 I = e−x cos x dx 0

Assuming that the above integral converges, use the Leibniz integral rule to evaluate it.

Give the answer in the form 4 k , where k is an exact constant.

∞ 2 You may use without proof e−x dx = 1 π . 2 0

π 2 MM1I , I = 4 16e

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Question 28 It is given that the following integral converges

∞ 1− cos ()1 x 6 dx .  x2 0

By introducing a parameter in the integrand and carrying out a suitable differentiation under the integral sign, show that

∞ 1 1− cos ()x π 6 dx = .  x2 12 0

proof

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Question 29 It is given that the following integral converges

1 2 x − 1  I =   dx .  ln x  0

By carrying out a suitable differentiation under the integral sign, show that

I =5ln3 − 3ln 3 .

V , MM1K , proof

Created by T. Madas

Question 30 It is given that the following integral converges

∞ −1t e2 ln t dt . 0

Evaluate the above integral by introducing a new parametric term in the integrand and carrying out a suitable differentiation under the integral sign.

You may assume that

∞  1 1 1   Γ′()()x =Γ x −++γ  −  . x kxk+   k=1 

2()−γ + ln 2

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Question 31

π 2 ln() 1+ cosα cos x I = dx . 0 cos x

By carrying out a suitable differentiation on I under the integral sign, show that

=1π2 − 1 α 2 . I 8 2

proof

Created by T. Madas

Question 32

π 2 ln() 1+ 3sin 2 x I = dx .  sin 2 x 0

By introducing a parameter a in the integrand and carrying out differentiation on I under the integral sign, show that

I = π .

proof

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Question 33

∞ n n e−x − e −()2x I = dx , n ∈» .  x 0

By carrying out a suitable differentiation on I under the integral sign, show that for all n ∈» ,

I = ln 2 .

proof

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Question 34

1π 2 exp(−1 tanx) − exp() − 3 tan x I = 3 dx .  sin 2 x 0

By carrying out a suitable differentiation on I under the integral sign, show that

= 1 ln3 . I 2

proof

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Question 35

π 2 1 dx , k . J = 2 2 ≠ 1 0 1+ k tan x

a) Use appropriate methods to find, in terms of k , a simplified expression for J .

π 2 arctan()k tan x I ()k= dx , k ≠ 1. 0 tan x

b) By carrying out a suitable differentiation on I under the integral sign, show that

π 2 xcot xdx = 1 π ln 2 . 2 0

c) Deduce the value of

π 2 ln() sin x dx . 0

π J = , − 1 π ln 2 2()k + 1 2

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Question 36 The integral function I( k ) is defined as

π I ()()k= ekcos x cos kxdx sin , k ∈» . 0

By carrying out a suitable differentiation on I under the integral sign, show that

π ecos x cos() sin x dx = π . 0

proof

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Question 37

π ln() 1+ cosα cos θ I = dθ . 0 cos θ

By carrying out a suitable differentiation on I under the integral sign, show that

π ln() 1+ cos θ π 2 dθ = . 0 cosθ 2

proof

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Question 38

π Ik()()≡ln 1 − kxdx cos , k <1. 0

By differentiating both sides of the above equation with respect to k , show that

  I() k=π ln1 1 − 1 − k 2 . 2 ( ) 

proof

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Question 39 Find the following inverse Laplace transform, by using differentiation under the integral sign.

  s  L−1 , a > 0 . 2  s2 a 2  ()+ 

  s  tsin at L−1 2  = s2 a 2 2a ()+ 

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Question 40 The integral I is defined in terms of the constants α and k , by

∞ 2 I()()α, k≡ e−α x cos kx dx , α > 0 . 0

By differentiating both sides of the above equation with respect to k , followed by integration by parts, show that

∞ 2  −α x2 π k e cos()kx dx = exp  −  . 0 4α 4 α 

You may assume without proof that

∞ 2 e−x dx = 1 π . 2 0

You may not use contour integration techniques in this question.

proof

Created by T. Madas

Question 41 It is given that the following integral converges

∞ −x e 1 1 −3x  3− + e  dx . 0 x x x 

By introducing a parameter λ and carrying out a suitable differentiation under the integral sign, show that

∞ −x e 1 1 −3x  3−+ e  dx =−+ 3ln256 . 0 x x x 

proof

Created by T. Madas

Question 42 The following integral is to be evaluated

π 2 2 2 22  lna cosθ+ b sin θ  d θ , 0 where a and b are distinct constants such that a+ b > 0 .

By carrying out a suitable differentiation under the integral sign, show that

π 2 2 2 22  a+ b  lna cosθ+ b sin θθπ  d = ln   . 0 2 

proof

Created by T. Madas

Question 43 It is given that

α + cos x  y = arcsin , 1+α cos x 

where α is a constant.

a) Show that

dy 1−α 2 = − . dx1+α cos x

The integral function I(α, x ) is defined as

π I()()α=ln 1 + α cos xdx . 0

b) By differentiating both sides of the above relationship with respect to α , show further that

I (1) = − π ln 2 .

proof

[ solution overleaf ]

Created by T. Madas

Question 44 By carrying out suitable differentiations on I under the integral sign, show that

∞ I =arccot()() 2x arccot 4 xdx = 1π ln 27 .  8() 4 0

V , proof

Created by T. Madas

Question 45

t 2 2  A() t≡ e−x dx  . 0 

By differentiating both sides of the above equation with respect to t , followed by the substitution x= ty , show that

∞ 2 e−x dx = 1 π . 2 0

proof

Created by T. Madas

Question 46

t 2 2  I ()t≡ e−i x dx  . 0 

By differentiating both sides of the above equation with respect to t , followed by the substitution x= ty , show that

∞ ∞ cosxdx2= sin xdx 2 = 1 2 π . ()() 4 0  0

proof

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INTEGRATION UNDER THE INTEGRAL SIGN

Created by T. Madas

Question 1 By integrating both sides of an appropriate integral relationship, with suitable limits, show that

1 xb− x a  b + 1  dx = ln   , 0 lnx a + 1  where b> a > 0 .

k x You may assume that for k > 0 , kx dx = + constant .  ln k

proof

Created by T. Madas

Question 2 By integrating both sides of an appropriate integral relationship, with suitable limits, show that

∞ 2 2 e−ax− e − bx dx b a , 2 =π − π 0 x where b> a > 0 .

∞ 2 You may assume that e−t dt = 1 π . 2 0

proof

Created by T. Madas

Question 3 The integral I is defined as

∞ I = ekx sin x dx . 0

where k is a constant.

a) Use a suitable method to show that

1 I = . k 2 +1

b) By integrating both sides of an appropriate integral relationship with respect to k , with suitable limits, show further that

∞ e−2x sin x dx = arccot 2 . 0 x

∞ sin x π You may assume that dx = . 0 x 2

proof

Created by T. Madas

Question 4 By integrating both sides of an appropriate integral relationship with respect to b , with suitable limits, show that

∞ esinh−x bx 1 1 + b  dx = ln   . 0 x2 1 − b 

proof

Created by T. Madas