9.3.2 Integrating Composite Exponential and Rational Functions by Substitution

Question 1

Integrating Composite Exponential and Rational Functions by Substitution

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

Evaluate.∫2x/x^2+5dx

Answer 3

ln∣x^2+5∣+C

Question 4

Evaluate the integral.

∫x^2e^x^3dx

Answer 4

1/3e^x^3+C

Question 5

Integrate.∫cosx⋅e^sinxdx

Answer 5

e^ sin x + C

Question 6

Integrate.∫dx/3x−2

Answer 6

1/3ln∣3x−2∣+C

Question 7

Integrate.∫e^x(1+e^x)^5dx

Answer 7

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

Question 8

Evaluate the integral.

∫e^√x/√x dx

Answer 8

2e^√x+C

Question 9

Evaluate.∫(lnx)^3/xdx

Answer 9

(lnx)^4/4+C

Question 10

Integrate.∫x+2/x^2+4x dx

Answer 10

ln∣x^2+4x∣/2+C