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.

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2

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Example 2

Thank you for your time