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The subsitution Method. Fatema Ahmed Alhajeri 201108382 #19. Previous information. an indefinite integral is a function ∫ f(x)dx= F(x) where F’(x) = f (x) which represents a particular antiderivitive of f, or an entire family of antiderivitives. Example 1. ∫ x √x dx - PowerPoint PPT Presentation
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The subsitution MethodFatema Ahmed Alhajeri
201108382#19
Previous information• an indefinite integral is a function ∫f(x)dx= F(x) where F’(x) = f(x) which represents a particular
antiderivitive of f, or an entire family of antiderivitives.
Example 1
• ∫ x √x dx• = ∫ x x1/2 dx• =∫ x (½ + 1) dx • = ∫x 3/2 dx • Therefore using the intermediate
integration rule:
• We can solve this previous equation:• F(x) = ∫ x3/2 dx• = (x5/2 ) + C
5/2
Therefore,• = F(x) = 2/5 x5/2+C
The Substitution Method• We need to use the substitution method in order
to convert difficult equations, to fit the intermediate general equations in order to solve.
• Example 2• f(x) = ∫ 2x √(1+x2) dx• = ∫ (1+x2 )1/2 2x dx
• Using substituion method we let u = 1+x2 since the derivitive of u (du) is = 2x and is present in the equation
u = 1 +x2
du= 2x dx
• We now change the equation in terms of “u” where it equals to:
• F (x)= ∫ (u)1/2 du• This equation fits the structure of:
• Thus,• ∫ (u)1/2 du • = ∫ (u)3/2 + C
3/2
• = 2/3 u3/2 + C
• Now, replacing the “u” with the variable x to retrieve the final answer:
• = 2/3 √(1+x2)3/2 + C
Math prject - Substitution rule 11
Chapter 5INTEGRALS
THE SUBSITUTION RULE5.5
Somaia Elsherif
Math prject - Substitution rule 12
The Substitution rule
• In general, Notice that each differentiation rule for functions provides a rule to find an antiderivative
• The substitution method is a rewriting method for integrals where we can extend the scope of these rules
• The idea behind substitution rule is to replace a relatively complicated integral by a simpler one
Math prject - Substitution rule 13
• The Chain rule for differentiation
• The chain rule implies the substitution rule
∫f’(g(x))g’(x) = f(g(x)) + C
• Substitution rule (according to text book)
∫ f(g(x))g’(x) dx = ∫ f(u) du
where u = g(x), then du = g’(x)
Summary of the substitution rule proof
Math prject - Substitution rule 14
The Substitution rule• If u=g(x) is a differentiable function whose range is an interval
I and f is continuous on I,Then ∫ f(g(x))g’(x) dx = ∫ f(u) du
• Notice that the rule was proved using the chain rule for differentiationThus, It is easy to remember it ! Only you have to think of dx and du in the previous formula as differentials
Math prject - Substitution rule 15
Example Find ∫ x³ cos (x⁴ + 2) dx.
• We made a substitution u = x⁴ + 2why? Because it’s differentiable function
• it’s differential is du = 4x³ dx, which, apart from the constant factor 4, occurs in the integral.
• Thus, using x³ dx = du/4 and the substitution rule we have
Math prject - Substitution rule 16
∫x³ cos (x⁴ + 2) dx
= ∫cos u . ¼ du
= ¼ ∫ cos u du
= ¼ sin u + C
• Notice that you have to return to the original variable x
= ¼ sin (x⁴ + 2) + C
Math prject - Substitution rule 17
Notice that
• The main challenge in using the Substitution rule is to think of an appropriate substitution
You have 2 ways to do it !• Choose u to be function in the integrand whose differential
occurs ( except for a constant factor )this is similar to the previous example case
Math prject - Substitution rule 18
If the first method didn’t work, Try this• Choose u to be some how complicated part of the integrand
( perhaps the inner function in a composite function )Finally !
• Finding the right substitution is a bit of art• It’s not unusual to guess wrong• If your first guess doesn’t work, just try another substitution
THE SUBSTITUTION RULE FOR DEFINITE INTEGRALS
Done by: Maha Mohd.Ibrahim.ID:201106851
Supervised by: Foud Al-muhannadi
General rule
• If g’ is continuous on [a,b] and f is continuous on the range of u=g(x), then:
=(
Example
• Evaluate:
To find the new limit of the integration we note that:When and when
• Therefore:
• Evaluate:
Method (1):
Method (2): ⇒
⇒
The integral now is:
We got the same answer!
Definite integral :symmetry
Gehad desouky 201104686
#26
Definite integral : symmetry
• We use the substitution by U to simplify the calculations of integrals of functions that are symmetric
• Example :
Substitution rule (symmetry)
• The main rule is : • If (F) is continues on [-a,a]
if (f) is even then [f(-x)=f(X)], then
If (F) is odd then [f(-x)=-f(X)], then
a
a
adxxfdxxf
0.)(2)(
a
adxXf 0)(
• Example 1, Evaluate by writing it As a sum of two integrals and interpreting
one of those integrals in terms of an area.
dxxx 2
2
24)3(
2
2
22
2
22
2
2
22
2
22
2
2
4344)3(
)()()()([
)434(4)3(
dxxdxxxdxxx
dxxgdxxfxgxf
dxxxxdxxx
b
a
b
a
b
a
04
0)(
)(4)(
]2,2[
4)(
4
)1
2
2
2
2
2
2
2
2
dxxx
dxxf
xfxxxf
xxxf
dxxx
a
a
2
4
]2,0[4
424
)()()(
4)(
4
)2
22
2
2
2
2
0
22
2
2
2
2
radius
xy
xy
dxxdxx
xsofxfxf
xxf
dxx
64)3(
6)2(304)3(
24
)2(2
)4
(242
)3
2
2
2
2
2
2
2
22
0
2
dxxx
dxxx
then
rdxx
9
18236
72836
172936
1)729(
36
1
)1(36
1)3(
36
16
)(
6
1
)(6
1
6
1
6
21
66
3
1
6
3
1
5
2
2
3
u
duu
dxxdu
dxxdu
xu
dxxx 531
0
2 )21(
Example 2
Thank you for your time