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)
Loading flashcards...
1
Q

Integrating Composite Exponential and Rational Functions by Substitution

A
  • 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.
2
Q

note

A
  • 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.
3
Q

Evaluate.∫2x/x^2+5dx

A

ln∣x^2+5∣+C

4
Q

Evaluate the integral.

∫x^2e^x^3dx

A

1/3e^x^3+C

5
Q

Integrate.∫cosx⋅e^sinxdx

A

e^ sin x + C

6
Q

Integrate.∫dx/3x−2

A

1/3ln∣3x−2∣+C

7
Q

Integrate.∫e^x(1+e^x)^5dx

A

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

8
Q

Evaluate the integral.

∫e^√x/√x dx

A

2e^√x+C

9
Q

Evaluate.∫(lnx)^3/xdx

A

(lnx)^4/4+C

10
Q

Integrate.∫x+2/x^2+4x dx

A

ln∣x^2+4x∣/2+C

Decks in AP Calculus AB Class (190):