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Section 2.6 Calculus 2

Strategies for Integration
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2.6 Strategies for Integration

2.6.1 Identifying Appropriate Integration Strategies

  • When encountering an integration problem, it’s useful to spot certain traits which can identify the best integration method to apply. The following list isn’t fool-proof, but checking these in order can help you identify likely techniques for integration.
    1. Use algebra to simplify the integrand first, if possible. Split up sums into separate integrals as necessary.
    2. Is the integrand a sum of constant multiples of known derivatives or polynomials? If so, simply integrate using Calculus 1 techniques.
    3. Is the integral of the form \(\int cf(g(x))g’(x)\,dx\): a nested function along with (a constant multiple of) its derivative? If so, use integration by substitution with \(u=g(x)\). (Section 2.1)
    4. Is the integrand a rational function (a fraction of two polynomials)? If so, try the method of partial fractions to expand the integrand algebraically. (Section 2.4)
    5. Does the integrand include only trigonometric functions? Use trigonometric identities to allow for a direct substitution. (Section 2.2)
    6. Does the integrand include expressions of the form \(a+bx^2\), \(a-bx^2\), or \(bx^2-a\)? Use the method of trigonometric substitution to simplify. (Section 2.3)
    7. Is the integrand the product of two functions? Integration by parts may produce a more manageable integral. (Section 2.5)
    8. At this point, check to make sure you didn’t miss a possibility above. Otherwise, you may need to use a combination of techniques from the above to proceed.
  • Example Find \(\int\sinh x\sqrt{1+\cosh x}\,dx\).
  • Example Find \(\int 2ze^{3z}\,dz\).
  • Example Find \(\int\sin^2 \theta+\cos^2 \theta\,d\theta\).
  • Example Find \(\int\frac{5x^2+12}{x^3+4x}\,dx\).
  • Example Find \(\int3\sec y\tan y-\frac{1}{1+y^2}\,dy\).
  • Example Find \(\int\frac{1}{\sqrt{4-9t^2}}\,dt\).
  • Example Find \(\int\sin^2 x\cos^3 x\,dx\).

Exercises for 2.6

For each of the following integrals in problems 1-7, first choose the most appropriate technique to begin integration. Then do the integration.

  1. \(\int(x^2-1)(x^2+1)\,dx\).
  2. \(\int\frac{1}{\sqrt{9+z^2}}\,dz\). (Recall \(\int\sec\theta\,d\theta=\ln|\sec\theta+\tan\theta|+C\).)
  3. \(\int 6y^2e^{y^3}\,dy\).
  4. \(\int 3x\sin(4x)\,dx\).
  5. \(\int\sec^3 \theta\tan^3 \theta\,d\theta\).
  6. \(\int\frac{5x-5}{x^2-3x-4}\,dx\).
  7. \(\int (4\sqrt{t}-3\tan(t)\sec(t))\,dt\).
  8. Find \(\int e^x\sqrt{1-e^{2x}}\,dx\) using a combination of different integration techniques. (Hint: \(\sin(2\theta)=2\sin\theta\cos\theta\).)
  9. Match each of these five integrals with the most appropriate of the integration technique to begin integration. Each technique from the below list will be used exactly once.
    1. \(\int\frac{4x}{x^2+3}\,dx\)
    2. \(\int\cos^3(x)\,dx\)
    3. \(\int\frac{5}{2x^2+8}\,dx\)
    4. \(\int\frac{x}{\csc(x)}\,dx\)
    5. \(\int\frac{4x^2+x+3}{x^3+3x^2}\,dx\)
  10. Match each of these five integrals with the most appropriate of the integration technique to begin integration. Each technique from the below list will be used exactly once.
    1. \(\int\sec^5(y)\tan^3(y)\,dy\)
    2. \(\int\frac{\sin(y)}{1-2\cos(y)}\,dy\)
    3. \(\int\frac{y^2+4y}{(y^2+4)(y+2)}\,dy\)
    4. \(\int\sqrt{4y^2-9}\,dy\)
    5. \(\int\cos(y)\sinh(y)\,dy\)

Choose from the following techniques for problems 9 & 10:

  • Integration by Substitution
  • Method of Partial Fractions
  • Trigonometric Identities
  • Trigonometric Substitution
  • Integration by Parts

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • Review of 5.5, 8.1, 8.2, 8.3, 8.4