Definition of a Definite Integral DEFINITE INTEGRALS

18 Definition of a Definite Integral DEFINITE INTEGRALS

Let f(x) be defined in an interval a ⬉ x ⬉ b. Divide the interval into n equal parts of length ⌬x = (b − a)/n. Then the definite integral of f(x) between x = a and x = b is defined as

18.1. b

a () f x dx = lim{ ( ) n fa ∆ x + fa ( ∆∆ x →∞ ) + x + fa ( + 2 ∆∆ x ) xx + ⋯ + fa ( + ( n − 1 ) ∆∆ x ) x ∫ }

The limit will certainly exist if f(x) is piecewise continuous. If f x d () = dx gx ( ), then by the fundamental theorem of the integral calculus the above definite integral can

be evaluated by using the result

18.2. () f x dx

∫ () a = ()

∫ () a ()

dx g x dx = gx a = gb − ga

If the interval is infinite or if f(x) has a singularity at some point in the interval, the definite integral is called an improper integral and can be defined by using appropriate limiting procedures. For example,

18.3. a () f x dx ∫ lim = b →∞ ∫ a () f x dx

f x dx lim ∫ () =

18.4. b ()

∫ f x dx a

18.5. a () f x dx = lim

f x dx

if is a singular point..

b 18.6. b

a () f x dx = ∫ lim ∈ → 0 ∫ a + fx () dx if is a singulaar point. a ∈

General Formulas Involving Definite Integrals

b b b 18.7. b

a {() ∫ fx ± gx () ± hx () ± ⋯ } dx = a () f x dx ± a () g x dx ± () ∫ h x dx ∫∫ ∫ a ± ⋯

b b 18.8. b

a () cf x dx = c a () f x dx where is any constant. ∫ c ∫

18.9. a

a () f x dx ∫ 0 =

b 18.10. a ∫

a () f x dx =− ∫ b () f x dx

18.11. ∫ a () f x dx = ∫ a () f x dx + c () ∫ f x dx

DEFINITE INTEGRALS

18.12. b

a () f x dx = ( b − afc )() where is between and c a b ∫ .

This is called the mean value theorem for definite integrals and is valid if f(x) is continuous in

a ⬉ x ⬉ b.

b 18.13. b

a ()() f x g x dx = fc ()

where is between and bb c ∫ a ∫ a g x dx

This is a generalization of 18.12 and is valid if f(x) and g(x) are continuous in a ⬉ x ⬉ b and g(x) ⭌ 0. Leibnitz’s Rules for Differentiation of Integrals

1 ∫ φ 11 () α d + φα −F (,) α 1 2 d α φα α 1 d α

Approximate Formulas for Definite Integrals In the following the interval from x = a to x = b is subdivided into n equal parts by the points

a =x 0 ,x 1 ,x 2 , …, x n –1 ,x n = b and we let y 0 = f(x 0 ), y 1 = f(x 1 ), y 2 = f(x 2 ), …, y n = f(x n ), h = (b – a)/n.

Rectangular formula:

18.15. b

a () f x dx ≈ hy ∫ ( 0 ++ y 1 y 2 + ⋯ + y n − 1 )

Trapezoidal formula:

18.16. a () f x dx ≈ ( y 0 + 2 y 1 + 2 y 2 + ⋯ + 2 y n − 1 + y n ∫ )

Simpson’s formula (or parabolic formula) for n even:

18.17. () f x dx

Definite Integrals Involving Rational or Irrational Expressions

dx

x 2 2 + = a 2 a ∞ x p − 1 dx

18.19. 0 π = , 0 p ∫ 1

x + a n = n sin[( m 1 )/]

π sin 18.21. m ∫ β

a dx

18.23. a 2 − x dx 2 ∫ π

DEFINITE INTEGRALS

a a m ++ 1 np [( m 1 )/ ] ( n p 1 )

18.24. 0 x m ( a n x n ) p dx

n Γ [( m + 1 ))/ n ++ p 1 ]

, 0 <+< m 1 ∫ nr

18.25. + 1 0 n n nr =

x dx m

() 1 − 1 r − π a m +− 1 nr Γ [( m )/ ]

n sin[ (( m + 1 ) / ]( π nr − 1 )! [( Γ m + 1 )/ n −+ r 1 ]

Definite Integrals Involving Trigonometric Functions All letters are considered positive unless otherwise indicated.

18.26. 0 sin mx sin nx dx ∫  = 

 0 mn , integers and m ≠ n

 π / 2 mn , intege rrs and m n =

  0 mn , integers and m ≠ n

18.27. 0 cos mx cos ∫ nx dx = 

 π / 2 mn , inntegers and m n =

0 mn , integers and m + n even

18.28. 0 sin mx cos ∫ nx dx = 

 2 mm /( 2 −− n 2 ) mn , integers and m + n odd

cos 2 x dx ∫ π 0 = ∫ 0 =

18.29. sin x dx

m 12 ,, ∫ … 0

18.30. sin x dx

cos x dx

= ∫ 0 = ii⋯

1352 ii⋯ m − 1

246 2 m 2 =

= m cos 2 + ∫ 1

π / 2 π / 2 246 ii⋯ 2 m

18.31. sin 2 m + 1 x dx

0 ∫ x dx 0 =

, m 12 ,, …

1352 ii⋯ m

π / 2 sin 21 p −

x cos 2 q − 1 x dx Γ ()() p Γ 18.32. q ∫ 0 =

∞ sin px

0 ∫ dx x =  0 p = 0

∫ 0 x dx =  π / 20 << p q

18.35. ∫ dx 0 x 2 = 

 π q / 2 p ⭌ q > 0

18.36. 0 2 dx ∫ π

∞ sin 2 px

∞ 1 cos px

x 2 dx

DEFINITE INTEGRALS

∞ cos px − cos qx

18.38. ∫ dx 0 ln

18.39. dx π ( q − p ∫ )

∞ cos px − cos qx

18.40. π − ∫ ma 0

∞ cos mx

x 2 2 dx =

18.41. dx π e − ∫ ma 0

∞ x sin mx

π ( 1 e − ma ∫ )

18.43. ∫ 0 a

18.44. ∫ 0 a =

+ b cos x

π / 2 dx

18.45. ∫ 0 a b =

cos ( / ) − 1 ba

+ cos x

dx

dx

( a + b sin ) x 2 = ∫ 00 ( a + b cos ) x 2 = ( a 2 b 232 ) − /

18.49. 0 2 2 2 , a < 1 , m = 012 ,,, ∫∫ …

cos mx dx

12 − a cos x + a =− 1 a

18.50. sin ax dx 2 cos ax dx 2 ∫ π

0 = ∫ 0 = 22 a

18.51. sin n

0 ∫ ax dx = n 1 n

na 1 / Γ ( / )sin , 2 n

18.52. 0 cos ax dx n = 1 / n Γ ( / ) cos 1 n π , n ∫ 1

∫ 0 dx x = 2

∞ sin x

18.54. 0 π p dx =

Γ ( )sin ( π /) 2 ∞ cos x

18.55. 0 dx

2 Γ ( ) cos ( p p π /) 2 <<

18.56. sin ax 2 0 b cos 2 bx dx π  ∫ cos = 

a − sin a 

22 a 

DEFINITE INTEGRALS

22 a + sin ∫   a a 

18.57. cos ax 2

0 cos 2 bx dx

cos

x 3 ∫ dx = 8

18.59. ∫ π 0 dx

x 4 = 3 ∞ tan x

18.60. dx ∫ π 0

1 + m tan x = 4 π / 2 x

π / 2 dx

18.64. dx ∫ π 0 = ln 2

∫ dx 0

18.65. − cos x

∞ cos x

− ∫ 1 x dx = γ

∞  1  dx

18.66. 0 2 − cos ∫ x = γ

 1 + x

 x

π 18.67. p ∫ dx

∞ tan − 1 px − − 1 tan qx

0 x ln = 2 q

Definite Integrals Involving Exponential Functions Some integrals contain Euler’s constant g = 0.5772156 . . . (see 1.3, page 3).

18.68. 0 e − ax cos ∫ bx dx =

18.69. e − ax

0 sin ∫ bx dx = a 2 + b 2

18.70. dx = tan − 1 ∫ b

0 ∫ dx x = ln a

0 e − ax dx

∫ bx dx =

18.73. e − ax 2

0 cos

DEFINITE INTEGRALS

π e ( − 4 ac )/ 4 a ∫ b

18.74. e − ( ax 2 ++ bx c )

dx

erfc

2 ∞ x 2 where erfc (p) =

e − dx

)/ 4 ∫ a

π ( b e 2 − 4 ac

18.75. e − ( ax 2 ++ bx c ) dx

Γ( n 1 18.76. ) xe dx ∫ +

− ax

− ax 2 m 18.77. )/ ] xe dx Γ[( + ∫ 12

0 = 2 a ( m + 12 )/

18.78. e − ( ax 2 + bx /) 2

18.79. ∫ 0 e x 1 1 2 + 2 2 + 3 2 + 4 2 + ⋯ =

18.80. 0 x dx = Γ( ) n 

e − + 1 1 n 2 n  + 3 n + 

For even n this can be summed in terms of Bernoulli numbers (see pages 142–143). ∞ x dx

0 e x += 1 1 2 − 2 2 + 3 2 − 4 2 + = 12

dx = Γ( ) n n − n + n − ⋯ ∫  0

18.82. x

e + 1  1 2 3 

For some positive integer values of n the series can be summed (see 23.10). ∞ sin mx

∫ coth 0

18.83. dx

18.84. − x  ∫ dx 0 e  1 +− x  x

0 x dx = 2 γ

18.86. ∫  0 dx

ln  + 18.87. p ∫ 0 = 

dx = cot − 1 0 a 2 a − ln ( a 2 1 ∫ )

∞ e − ax ( 1 cos ) x

DEFINITE INTEGRALS

Definite Integrals Involving Logarithmic Functions

1 ()! 1 n n

18.90. 0 x m (ln ) x dx n

+ 1 + 1 m >− 1 , n = 012 ,,, …

If n ≠ 0, 1, 2, … replace n! by Γ(n + 1).

18.91. 0 1 x ∫ dx =−

1 ln x

18.92. 0 1 − x ∫ dx =− 6

1 ln x

18.93. ∫ + 0 x dx = 12

1 ln ( 1 x )

18.94. ∫ 0 x dx =− 6

1 ln ( 1 − x )

18.95. ln ln ( x 1 + x dx )

∫ 0 =− 22 ln 2 − 12

18.96. ln ln ( x 1

0 − x dx ∫ ) =− 2 6

18.97. 0 1 x dx =− π 2 csc p π cot p π 0 << p ∫ 1

18.99. − 0 x e ln ∫ x dx =− γ

22 ∫ ln )

e − x ln x dx =−

ln  + ∫ 

∫ ln ∫ =− 2

(ln ) ∫ 2

0 x dx = 2 + 24

ln ∫ 2

x ln sin x dx =−

∫ 21 = ln −

sin ln sin

x dx

0 + b cos ) x dx = 2 π a + a 2 − b 2 ∫ ) ∫

ln ( sin )

0 a + b x dx =

ln ( a ln (

= π 0 ln

ln( a + b cos ) x dx

DEFINITE INTEGRALS

= ln ∫ 2 8

dx = {(cos − 1 a ) 2 − ( ccos − 1 b )} ∫ 2

0 sec ln x

 1 + a cos x 

 sin a sin 2 a sin 3 18.111. a ln 2 sin dx =− ⋯ ∫ 

See also 18.102.

Definite Integrals Involving Hyperbolic Functions

= a tanh ∫ π

∞ sin ax

sinh bx

0 cosh bx =

0 sinh ax = 4 a 2

sinh ax 2 a + { 1 n + 1 + 2 n + 1 ++ 3 n + 1 + }

x dx n

If n is an odd positive integer, the series can be summed. ∞ sinh ax

dx π csc ∫ π 0

e bx =

∞ sinh ax

18.117. ∫ 0 bx e dx = − cot

Miscellaneous Definite Integrals

() f ax () f bx

0 dx = {() f 0 −∞ f ∫ ( )}ln x a

∞ fx () −∞ f ()

This is called Frullani’s integral. It holds if f ′(x) is continuous and ∫ 0 dx converges.

1 dx 1 1 1

18.120. a ( a x ) m − 1 ( a x ) n − 1 dx () 2 a + mn − = +− 1 Γ ()() m Γ ∫ n

Section V: Differential Equations and Vector Analysis

BASIC DIFFERENTIAL EQUATIONS

and SOLUTIONS

DIFFERENTIAL EQUATION SOLUTION

19.1. Separation of variables

∫ dx +

∫ dy = c

fx 2 ()

gy 1 ()

19.2. Linear first order equation dy +

pxy () = Qx ()

ye ∫ Pdx = Qe ∫ Pdx dx ∫ + c

dx

19.3. Bernoulli’s equation dy

( 1 − n ) Pdx

( 1 − n ) + Pdx Pxy () = Qxy () ␷e ∫ =− ( 1 n ) Qe ∫ dx ∫ + c

dx

where y

=y 1 −n . If n = 1, the solution is

ln y = ( Q ∫ − P dx ) + c

19.4. Exact equation M (x, y)dx + N(x, y)dy = 0

⎛ N ∂ M x dy ⎞ ∫ c

Mx

∂ y ∫ ⎠⎟

where ∂M/∂y = ∂N/∂x. where ∂x indicates that the integration is to be per-

formed with respect to x keeping y constant.

19.5 Homogeneous equation

∫ F () ␷ − ␷

⎝⎜ x ⎠⎟

where y = y/x. If F(y) = y, the solution is y = cx.

BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS

y F(xy) dx + x G(xy) dy = 0

G () ␷ d ␷

␷ {() G ␷ − F ( )} ␷ + where y = xy. If G(y) = F(y), the solution is xy = c.

ln x =

19.7. Linear, homogeneous second order Let m 1 ,m 2 be the roots of m 2 + am + b = 0. Then there equation

are 3 cases.

Case 1. m 1 ,m 2 real and distinct:

Case 2. m 1 ,m 2 real and equal: y mx = ce 1 a mx , b are real constants.

1 + cxe 1 2 Case 3. m 1 = p + qi, m 2 = p − qi: y = e px ( cos c 1 qx + c 2 sin qx )

where p = −a/2, q = b − a 2 /. 4

19.8. Linear, nonhomogeneous second There are 3 cases corresponding to those of entry 19.7 order equation

a, b are real constants.

e mx 2 − + mx

m 22 − m 1 ∫

xe mx 1 − + mx e 1 R x dx ∫ ()

e mx xe − − m 1 11 x R x dx ∫ ()

e Rx ( )sin qx dx

BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS

19.9. Euler or Cauchy equation Putting x =e t , the equation becomes

and can then be solved as in entries 19.7 and 19.8 above.

19.10. Bessel’s equation dy 2 2 dy

See 27.1 to 27.15.

19.11. Transformed Bessel’s equation

where q

= 2 p − β. 2

19.12. Legendre’s equation dy 2 dy

( 1 − x 2 ) 2 − 2 x + nn ( + 1 ) y = 0 y = cPx 1 n () + cQx 2 n () dx

dx

See 28.1 to 28.48.