9.3.2 Integrating Composite Exponential and Rational Functions by Substitution Flashcards Preview

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

Flashcards in 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution Deck (10):
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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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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.

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Evaluate.∫2x/x^2+5dx

ln∣x^2+5∣+C

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Evaluate the integral.
∫x^2e^x^3dx

1/3e^x^3+C

5

Integrate.∫cosx⋅e^sinxdx

e^ sin x + C

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Integrate.∫dx/3x−2

1/3ln∣3x−2∣+C

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Integrate.∫e^x(1+e^x)^5dx

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

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Evaluate the integral.
∫e^√x/√x dx

2e^√x+C

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Evaluate.∫(lnx)^3/xdx

(lnx)^4/4+C

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Integrate.∫x+2/x^2+4x dx

ln∣x^2+4x∣/2+C

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