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AP Calculus AB: 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution

Mathematics10 CardsCreated 3 months ago

This section teaches how to apply integration by substitution to composite exponential and rational functions. It emphasizes choosing a suitable substitution u, often the inner function or denominator, and adjusting for constants when du isn’t a perfect match. The content provides strategic guidance and worked examples, highlighting the importance of expressing final answers in terms of the original variable.

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Integrating Composite Exponential and Rational Functions by Substitution

  • Integration by substitution is a technique for finding the antiderivative of a composite function. A composite function is a function that results from first applying one function, then another.

  • You may need to experiment with several choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewhere in the integrand.

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Key Terms

Term
Definition

Integrating Composite Exponential and Rational Functions by Substitution

  • Integration by substitution is a technique for finding the antiderivative of a composite function. A composite function is a function that ...

note

  • The first step when integrating by substitution is to identify the expression that you will replace with u. There will often be many candid...

Evaluate.∫2x/x^2+5dx

ln∣x^2+5∣+C

Evaluate the integral.

∫x^2e^x^3dx

1/3e^x^3+C

Integrate.∫cosx⋅e^sinxdx

e^ sin x + C

Integrate.∫dx/3x−2

1/3ln∣3x−2∣+C

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AP Calculus AB: 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution
TermDefinition

Integrating Composite Exponential and Rational Functions by Substitution

  • Integration by substitution is a technique for finding the antiderivative of a composite function. A composite function is a function that results from first applying one function, then another.

  • You may need to experiment with several choices for u when using integration by substitution. A good choice is one whose derivative is expressed elsewhere in the integrand.

note

  • The first step when integrating by substitution is to identify the expression that you will replace with u. There will often be many candidates for u. A good strategy is to pick one and test it. In this case, differentiating the expression in the first box produces 40x^3 + 4, but there is no other cubic in the integrand.

  • Choosing the expression in the second box and differentiating gives you an expression with a fourth power and a first power. The exact expression is not in the integrand. However, it can be multiplied by 2 to give the expression in the first box.

  • Once you have determined the expression for u, the integrand should be simple to evaluate. Remember to replace u with its expression in terms of x.

  • In the case of a rational integrand, the best choice for u may be the denominator. In this example, du then appears in the numerator.

  • Replace the expressions in terms of x with the corresponding u- and du-expressions.

  • The integral of du/u is ln|u| + C.

  • You have not finished the technique until you have your result expressed in terms of x.

Evaluate.∫2x/x^2+5dx

ln∣x^2+5∣+C

Evaluate the integral.

∫x^2e^x^3dx

1/3e^x^3+C

Integrate.∫cosx⋅e^sinxdx

e^ sin x + C

Integrate.∫dx/3x−2

1/3ln∣3x−2∣+C

Integrate.∫e^x(1+e^x)^5dx

(1+e^x)^6/6+C

Evaluate the integral.

∫e^√x/√x dx

2e^√x+C

Evaluate.∫(lnx)^3/xdx

(lnx)^4/4+C

Integrate.∫x+2/x^2+4x dx

ln∣x^2+4x∣/2+C