Integration Using Substitution

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IntegrationUsing Substitution

• Pauline Aguayo
• Brent Rivas
• Adam Ortiz
• Matthew Gazmen
• Araceli Resendiz
• Katie Jensen

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INTEGRAL anti-derivative
INTEGRATING
Integrating is finding the integral of something

• The method of integration by substitution for
  indefinite integrals involves changing a
  variable( say x ) of an integrand to another
  variable( u ).

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Five Steps of Integrating by Substitution 1) Let
u the inner part of a function. 2) Find du 3)
Re-write the integral in terms of u (and change
the limits if necessary.) 4) Evaluate the
integral in terms of u 5) Replace u with the
original g(x)
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Formulas

• Properties of Indefinite Integrals
• ?kf (x)dx k?f (x)dx for any constant k
• ?(f(x)g(x))dx ?f(x)dx ? g(x)dx
• Power Formulas
• ?undu (u(n1))/(n1) C when C? -1
• ?u-1du ?1/u du ln(u) C

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Formulas(continued)
Trigonometric Functions

• ?(csc(u))2du -cot(u) C
• ?sec(u)tan(u)du sec(u) C
• ?csc(u)cot(u)du -csc(u) C

• ?cos(u)du sin(u) C
• ?sin(u)du-cos(u) C
• ?(sec(u))2du tan(u) C

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Formulas(continued)

• Exponential and Logarithmic
• ?eudu eu C
• ?ln(u)du uln(u)- u C
• ?loga u du ?(ln(u)/In(a))du (uln(u)- u)/
  (ln(a)) C
• ?au du (au)/ (ln(a)) C

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Example1
?sin(3x5)dx

• Step 1 Let u(3x5)

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?sin(3x5)dx
Step 1 Let u(3x5)
u 3x5 du 3 dx

• Step 2 Find du

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?sin(3x5)dx
Step 1 Let u(3x5)
Step 2 Find du du3

• Step 3 Find dx

• du 3 dx
• du 3 dx
• 3
• 1 du dx
• 3 m

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?sin(3x5)dx
Step 1 Let u(3x5)
Step 2 Find du du3
Step 3 Find dx dx du
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• Step 4 Substitute

?sin(u) dx ?sin(u) du
3 ??sin(u)du ??sin(3x5)du
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?sin(3x5)dx
Step 1 Let u(3x5)
Step 2 Find du du3
Step 3 Find dx dx du
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• Step 4 Substitute ??sin(3x5)du

Step 5 Integrate
??sin(3x5)du ??(-cos(3x5)) C
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?sin x ecos x dx
Example2

• -?(-sin x)ecos x dx
• -?ecos x (-sin x)dx
• -?eu du
• -eu C
• -ecos x C

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?x²v(5 2x³)dx
Example3

• ? (5 2x³)½ x² dx
• 1/6 ? (5 2x³)½ 6x² dx
• 1/6 ? u½ du
• 1/6(?)u3/2 C
• 1/9(5 2x³)3/2 C

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Example4

• ? x cos (2x2) dx

• ? x x cosu x dx
• ? x x cosu x 1/4x du
• ¼ ? cosu x du
• ¼ sinu C
• ¼ sin (2x2) C

• First Step let u 2x2
• du 4x
• 1/4x du dx
• Second Step Replace 2x2 with u
• Third Step Multiply x x cosu x 1/4x du and then
  you cancel the x.
• Fourth Step Change cos to sin

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Thank You!