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Active Calculus

Matthew Boelkins

Section 9.3 Integrating Rational Functions

Subsection 9.3.1 Preview Activity

Question 9.3.1.

For each of the indefinite integrals below, decide whether the integral can be evaluated: immediately because it’s a basic integral, using u-substitution, integration by parts, with multiple techniques, or if none of those will work.
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∫11+x2dx
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∫x1+x2dx
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∫2x+31+x2dx
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∫ex1+(ex)2dx
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∫1x2−1dx
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Hint.
Note that 2x+31+x2 can be rewritten as 2x1+x2+31+x2.
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Question 9.3.2.

First, note that we can use algebra to combine −8x+3+2x−3 into one fraction by finding a common denominator.
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−8x+3+2x−3=−8x+3⋅x−3x−3+2x−3⋅x+3x+3=30−6xx2−9
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Also note that we know how to integrate
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∫−8x+3dx= and
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∫2x−3dx=
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Which means we can integrate both sides of −8x+3+2x−3=30−6xx2−9 to get
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∫30−6xx2−9dx=
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Hint.
Remember that you really should use absolute value instead of parentheses when you integrate to get ln⁡. Sometimes WeBWorK doesn’t care about the difference, but sometimes it does. For example, ∫1x+11dx=ln⁡|x+11|+C, not ln⁡(x+11)+C.
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Question 9.3.3.

Find the quotient and remainder using long division for
x5−x4+x3−x2+3x−12x−1.
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The quotient is
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The remainder is
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Question 9.3.4.

Perform the indicated division and write the quotient and remainder in the provided blanks.
3x2−x−6x−1
Answer: + /(x−1)
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Exercises 9.3.2 Exercises

1.

Which of the following is the correct form of the partial fraction decomposition of x−1x3+x2? You may select more than one answer.
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  1. Ax+Bx+Cx+Dx2+Ex+1
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  2. Ax+Bx2+Cx+1
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  3. Ax+Bx+Cx+Dx2+Ex+Fx+1
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  4. Ax+Bx+Cx+Dx2+Ex+Fx+1
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2.

Which of the following is the correct form of the partial fraction decomposition of x−1x3+x? You may select more than one answer.
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  1. Ax+Bx+Cx+Dx2+1
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  2. Ax+Bx2+1
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  3. Ax+Bx+Cx2+1
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  4. Ax+Bx+Cx2+1
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3.

What is the correct form of the partial fraction decomposition for the following integral?
∫x−10x3+4x2−45xdx
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  • A. ∫(Ax+Bx−5+Cx+9)dx
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  • B. ∫(Ax+Bx+Cx−10+Dx+Ex+5)dx
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  • C. There is no partial fraction decomposition because the denominator does not factor.
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  • D. ∫(Ax−10+Bx−9+Cx+5)dx
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  • E. There is no partial fraction decomposition yet because there is cancellation.
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  • F. There is no partial fraction decomposition yet because long division must be done first.
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  • G. ∫(Ax+Bx−10+Cx+5)dx
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  • H. ∫(Ax+Bx+Cx+Dx−5+Ex+Fx+9)dx
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4.

What is the correct form of the partial fraction decomposition for the following integral?
∫10(x8+6)(x−7)(x2−1)2(x2+7)2dx
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  • A. There is no partial fraction decomposition because the denominator does not factor.
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  • B. ∫(Ax−7+Bx+1+C(x+1)2+Dx−1+E(x−1)2+Fx+Gx2+7)dx
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  • C. ∫(Ax−7+Bx+C(x2−1)2+Dx+E(x2+7)2)dx
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  • D. ∫(Ax−7+Bx+Cx2−1+Cx+D(x2−1)2+Ex+Fx2+7+Gx+H(x2+7)2)dx
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  • E. ∫(Ax−7+B(x2−1)2+C(x2+7)2)dx
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  • F. There is no partial fraction decomposition yet because long division must be done first.
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  • G. There is no partial fraction decomposition yet because there is cancellation.
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5.

Consider the indefinite integral ∫6x3+2x2−49x−21x2−9dx
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Then the integrand decomposes into the form
ax+b+cx−3+dx+3
where
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a =
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b =
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c =
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d =
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Integrating term by term, we obtain that
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∫6x3+2x2−49x−21x2−9dx=
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+C
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6.

Evaluate the integral
∫10(x+a)(x+b)dx
for the cases where a=b and where a≠b.
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Note: For the case where a=b, use only a in your answer. Also, use an upper-case "C" for the constant of integration.
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If a=b:
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If a≠b:
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7.

Consider the integral
∫x21−3x14+6x7−16(x3−6x2+5x)3(x4−625)2dx
Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand. A1,A2,A3… and B1,B2,B3,… denote constants.
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You must get all of the answers correct to receive credit.
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8.

Calculate the integral below by partial fractions and by using the indicated substitution. Be sure that you can show how the results you obtain are the same.
∫2xx2−4dx
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First, rewrite this with partial fractions:
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∫2xx2−4dx=∫ dx+∫ dx= + +C.
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(Note that you should not include the +C in your entered answer, as it has been provided at the end of the expression.)
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Next, use the substitution w=x2−4 to find the integral:
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∫2xx2−4dx=∫ dw= +C= +C.
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(For the second answer blank, give your antiderivative in terms of the variable w. Again, note that you should not include the +C in your answer.)
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