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Integrals
Definitions Definite Integral: Suppose f (^) ( x (^) )is continuous
on (^) [ a b , ]. Divide (^) [ a b , (^) ]into n subintervals of
width ∆ x and choose x i *from each interval.
Then (^) ( ) (^) ( *) 1
lim (^) i
b n a n (^) i f x dx f x x →∞ (^) = ∫ =^ ∑ ∆.
Anti-Derivative : An anti-derivative of f (^) ( x ) is a function, F (^) ( x (^) ), such that F ′ (^) ( x (^) ) = f (^) ( x ). Indefinite Integral : (^) ∫ f (^) ( x dx ) = F (^) ( x (^) )+ c where F (^) ( x (^) )is an anti-derivative of f (^) ( x (^) ).
Fundamental Theorem of Calculus
Part I : If f (^) ( x (^) )is continuous on (^) [ a b , ] then
( ) ( )
x a g x = (^) ∫ f t dt is also continuous on (^) [ a b , ]
and (^) ( ) ( ) ( )
x a
d g x f t dt f x dx
′ (^) = (^) ∫ =.
Part II : f (^) ( x (^) )is continuous on[ a b , ] , F (^) ( x (^) ) is
an anti-derivative of f (^) ( x (^) )( i.e. F (^) ( x (^) ) = (^) ∫ f (^) ( x dx ) )
then (^) ( ) ( ) ( )
b ∫ (^) a f^ x dx^ =^ F b^ − F^ a.
Variants of Part I :
( )
( ) ( ) ( )
u x a
d f t dt u x f u x dx ∫ =^ ′ ^
( ) ( ) (^ )^ (^ )
b v x
d f t dt v x f v x dx ∫ = −^ ′ ^
( )^ ( )
( ) ( ) [ ( )]^ ( ) [ ( )]
u x v x u x^ v x
d f t dt u x f v x f dx ∫ =^ ′^ − ′
Properties ∫ f^^ (^ x^ )^ ±^ g^ (^ x dx )^^ =^ ∫ f^ (^ x dx )^^ ±∫ g^ (^ x dx )
( ) ( ) ( ) ( )
b b b ∫ a (^) f^ x^ ±^ g^ x dx^ =^ ∫ (^) a f^ x dx^ ±∫ ag^ x dx
( ) 0
a ∫ a f^ x dx^ =
( ) ( )
b a ∫ a (^) f^ x dx^ = −∫ b f^ x dx
∫ cf^^ (^ x dx )^^ = c^ ∫ f^ (^ x dx ) ,^ c^ is a constant
( ) ( )
b b ∫ (^) a cf^ x dx^ = c^ ∫ a f^ x dx ,^ c^ is a constant
( )
b ∫ a c dx^ =^ c b^ − a
( ) ( )
b b ∫ (^) a f^ x dx^ ≤∫ a f^ x^ dx
( ) ( ) ( )
b c b ∫ a (^) f^ x dx^ =^ ∫ a (^) f^ x dx^ +∫ c f^ x dx for any value of^ c.
If f (^) ( x (^) ) ≥ g (^) ( x )on a ≤ x ≤ b then (^) ( ) ( )
b b ∫ (^) a f^ x dx^ ≥∫ ag^ x dx
If f (^) ( x (^) ) ≥ 0 on a ≤ x ≤ b then (^) ( ) 0
b ∫ a f^ x dx^ ≥
If m ≤ f (^) ( x (^) )≤ M on a ≤ x ≤ b then (^) ( ) ( ) ( )
b m b − a ≤ (^) ∫ a f x dx ≤ M b − a
Common Integrals ∫^ k dx^ =^ k x^ + c 1 1 1 ,^1
n n x dx (^) n x c n
∫ =^ + +^ ≠ − (^1 1) ln x dx (^) xdx x c
∫ ∫ (^1 1) ln ∫ a x + b dx^ =^ a ax^ +^ b^ + c
∫ ln^ u du^ =^ u^ ln^ (^ u^ )−^ u^ + c u (^) du = u + c ∫ e^ e
∫^ cos^ u du^ =^ sin u^ + c
∫ sin^ u du^ = −^ cos u^ + c sec^2 u du = tan u + c ∫
∫ sec^ u^ tan^ u du^ =^ sec u^ + c
∫ csc^ u^ cot^ udu^ = −^ csc u^ + c csc^2 u du = − cot u + c ∫
∫ tan^ u du^ =^ ln sec u^ + c
∫sec^ u du^ =^ ln sec^ u^ +^ tan u^ + c
∫ a^2 +^1 u^2 du^ =^ a^1^ tan^ −^1 (^ ua )+ c
2 1 2 sin^1 (^ ua ) a u
du − c − ∫ =^ +
Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class.
u Substitution : The substitution u = g (^) ( x )will convert (^) ( ( )) ( ) ( ) ( )
b g b ( ) a f^ g^ x^ g^ x dx^ g a f^ u^ du ∫ ′^ =∫ using du = g ′ ( x dx ). For indefinite integrals drop the limits of integration.
Ex.^2 ( 3 )
2 1 ∫^5 x^ cos x^ dx (^3 2 2 ) u = x ⇒ du = 3 x dx ⇒ x dx = 3 du x = 1 ⇒ u = 13 = 1 :: x = 2 ⇒ u = 23 = 8
( ) (^ )
( ) ( ( ) ( ))
(^2 2 38 ) 1 13 5 8 5 3 1 3
5 cos cos
sin sin 8 sin 1
x x dx u du
u
∫ ∫
Integration by Parts : (^) ∫ u dv = uv −∫ v du and
b (^) b b ∫ (^) a u dv^ =^ uv^ a −∫ (^) av du. Choose^ u^ and^ dv^ from
integral and compute du by differentiating u and compute v using v = (^) ∫ dv.
Ex. (^) ∫ x e − xdx
u = x dv = e −^^ x^ ⇒ du = dx v = − e − x x −^^ x^ dx = − x −^ x^ + −^ x^ dx = − x − x^^ − − x + c ∫ e^ e^ ∫ e^ e^ e
Ex.
5 3 ∫ ln^ x dx ln 1 u = x dv = dx ⇒ du = (^) xdx v = x
( ( ) )
( ) ( )
5 5 5 5 3 ln^ ln^3 3 ln 3 5ln 5 3ln 3 2
x dx = x x − dx = x x − x
= − −
∫ ∫
Products and (some) Quotients of Trig Functions
For (^) ∫ sin n^ x cos mx dx we have the following :
1. n odd. Strip 1 sine out and convert rest to
cosines using sin^2 x = 1 − cos^2 x , then use the substitution u = cos x.
2. m odd. Strip 1 cosine out and convert rest
to sines using cos^2 x = 1 − sin^2 x , then use the substitution u = sin x.
3. n and m both odd. Use either 1. or 2. 4. n and m both even. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated.
For (^) ∫ tan n^ x sec mx dx we have the following :
1. n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using tan 2 x = sec^2 x − 1 , then use the substitution u = sec x. 2. m even. Strip 2 secants out and convert rest to tangents using sec^2 x = 1 + tan^2 x , then use the substitution u = tan x. 3. n odd and m even. Use either 1. or 2. 4. n even and m odd. Each integral will be dealt with differently. Trig Formulas : sin 2( x (^) ) = 2sin (^) ( x (^) ) cos( x ), cos^2 ( x (^) ) = (^12) ( 1 + cos 2( x )), sin (^2) ( x (^) ) = (^12) ( 1 −cos 2( x ))
Ex. (^) ∫tan^3 x sec^5 x dx
( )
( ) ( )
3 5 2 4
2 4
2 4
1 7 1 5 7 5
tan sec tan sec tan sec
sec 1 sec tan sec
1 sec
sec sec
x xdx x x x xdx
x x x xdx
u u du u x
x x c
∫ ∫
∫
∫
Ex.
5 3
sin cos
x x ∫ dx
( )
1 2 1 2 2 2
5 4 2 2 3 3 3 2 2 3 2 2 (^2 ) 3 3
sin sin sin (sin^ )sin cos cos cos (1 cos )sin cos (1 ) (^) 1 2
cos
sec 2 ln cos cos
x x x x^ x x x x x x x u (^) u u u u
dx dx dx
dx u x
du du
x x x c
−
− (^) − +
∫ ∫ ∫
∫
∫ ∫
Applications of Integrals
Net Area : (^) ( )
b a ∫ f^ x dx represents the net area between^ f^ (^ x^ )and the
x -axis with area above x -axis positive and area below x -axis negative.
Area Between Curves : The general formulas for the two main cases for each are,
( ) upper function^ lower function
b y f x A (^) a dx = ⇒ = (^) ∫ ^ ^ −^ & (^) ( ) right function left function
d x f y A (^) c dy = ⇒ = (^) ∫ ^ ^ −^
If the curves intersect then the area of each portion must be found individually. Here are some sketches of a couple possible situations and formulas for a couple of possible cases.
( ) ( )
b a A = (^) ∫ f x − g x dx (^ )^ (^ )
d c A = (^) ∫ f y − g y dy^ c^ ( ) ( ) b ( ) ( ) A = (^) ∫ (^) a f x − g x dx + (^) ∫ cg x − f x dx
Volumes of Revolution : The two main formulas are V = (^) ∫ A x dx ( ) and V = (^) ∫ A y dy ( ). Here is
some general information about each method of computing and some examples. Rings Cylinders A = π( ( outer radius ) 2 − ( inner radius)^2 ) A = 2 π( radius ) ( width / height) Limits: x / y of right/bot ring to x / y of left/top ring Limits : x / y of inner cyl. to x / y of outer cyl.
Horz. Axis use f (^) ( x (^) ),
g ( x ), A x ( )and dx.
Vert. Axis use f (^) ( y (^) ), g ( y ), A y ( )and dy.
Horz. Axis use f (^) ( y (^) ), g ( y ), A y ( )and dy.
Vert. Axis use f (^) ( x (^) ), g ( x ), A x ( ) and dx.
Ex. Axis : y = a > 0 Ex. Axis : y = a ≤ 0 Ex. Axis : y = a > 0 Ex. Axis : y = a ≤ 0
outer radius : a − f (^) ( x )
inner radius : a − g (^) ( x )
outer radius: a + g (^) ( x ) inner radius: a + f (^) ( x )
radius : a − y width : f (^) ( y (^) ) − g (^) ( y )
radius : a + y width : f (^) ( y (^) ) − g (^) ( y )
These are only a few cases for horizontal axis of rotation. If axis of rotation is the x -axis use the y = a ≤ 0 case with a = 0. For vertical axis of rotation ( x = a > 0 and x = a ≤ 0 ) interchange x and
y to get appropriate formulas.
Work : If a force of F ( x )moves an object
in a ≤ x ≤ b , the work done is ( )
b
W = ∫ aF x dx
Average Function Value : The average value
of f ( x )on a ≤ x ≤ b is 1 ( )
b
f avg = b − a ∫ a f x dx
Arc Length Surface Area : Note that this is often a Calc II topic. The three basic formulas are, b a
L = ∫ ds 2
b a
SA = ∫ π y ds (rotate about x -axis) 2
b a
SA = ∫ π x ds (rotate about y -axis)
where ds is dependent upon the form of the function being worked with as follows.
( ) (^ )
2 1 if , dy ds = + (^) dx dx y = f x a ≤ x ≤ b
( ) (^ )
2 ds = 1 + dxdy dy if x = f y , a ≤ y ≤ b
2 2 if , , dx^ dy ds = (^) dt + (^) dt dt x = f t y = g t a ≤ t ≤ b
2 2
ds = r + ddr θ d θ if r = f θ , a ≤ θ≤ b
With surface area you may have to substitute in for the x or y depending on your choice of ds to match the differential in the ds. With parametric and polar you will always need to substitute.
Improper Integral An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. Integral is called convergent if the limit exists and has a finite value and divergent if the limit doesn’t exist or has infinite value. This is typically a Calc II topic.
Infinite Limit
1. ( ) lim ( )
t a t a f x dx f x dx →∞
∞
∫ = ∫ 2.^ (^ )^ lim (^ )
b b t t f x dx f x dx − ∞ →−∞
c c f x dx f x dx f x dx − −
∞ ∞ ∞ ∞
∫ =^ ∫ +∫ provided BOTH integrals are convergent.
Discontinuous Integrand
1. Discont. at a : ( ) lim ( )
b b
∫ a f^ x dx^ = t → a +∫ t f^ x dx 2.^ Discont. at^ b^ :^ (^ )^ lim (^ )
b t
∫ a f^ x dx^ = t → b −∫ af^ x dx
3. Discontinuity at a < c < b : ( ) ( ) ( )
b c b
∫ a f^ x dx^ =^ ∫ a f^ x dx^ +∫ c f^ x dx provided both are convergent.
Comparison Test for Improper Integrals : If f ( x ) ≥ g ( x ) ≥ 0 on [ a , ∞ )then,
1. If a f ( x dx )
∞
∫ conv. then^ a g^ (^ x dx )
∞
∫ conv.^ 2.^ If^ a g^ (^ x dx )
∞
∫ divg. then^ a f^ (^ x dx )
∞
∫ divg.
Useful fact : If a > 0 then (^) a^1 p x dx
∞
∫ converges if^ p^ >^1 and diverges for^ p^ ≤^1.
Approximating Definite Integrals
For given integral ( )
b a
∫ f^ x dx and a^ n^ (must be even for Simpson’s Rule) define^ ∆ x^^ =^ b^ − na and
divide [ a b , ] into n subintervals [ x 0 , x 1 ], [ x 1 , x 2 ], … , [ xn − 1 , xn ]with x 0 = a and xn = b then,
Midpoint Rule : ( ) ( 1 * ) ( * 2 ) ( *)
b
∫ a f^ x dx^ ≈ ∆ x^^ ^ f^ x^ +^ f^ x^ +^ ^ + f^ xn ,^
x i is midpoint [ xi − 1 , xi ]
Trapezoid Rule : ( ) ( 0 ) 2 ( 1 ) 2 ( 2 ) 2 ( 1 ) ( )
b a n^ n
x f x dx f x f x f x f x (^) − f x
∫ ≈^ ^ +^ + +^ +^ +^ +
Simpson’s Rule : ( ) ( 0 ) 4 ( 1 ) 2 ( 2 ) 2 ( 2 ) 4 ( 1 ) ( )
b a n^ n^ n
x f x dx f x f x f x f x (^) − f x (^) − f x
∫ ≈^ ^ +^ +^ +^ +^ +^ +
