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Trigonometric Substitution Suggestions

When using substitution to solve integrals, it is often obvious that trigonometric substitutions may work when trigonometric functions occur in the integrand. However, some integrands which appear to be algebraic can be simplified with a trigonometric substitution.

Trigonometric Substitutions

These substitutions use the substitution technique of integration. That means substitutions must be done to the integrand, to the differential, and in the case of definite integrals, to the limits of integration. They also must be undone after the integration in order to provide a sensible answer.

Suggested Substitutions
IntegrandSubstitutionDifferential Substitution
{$ \displaystyle \cos\theta d\theta $}{$ \displaystyle u=\cos\theta $}{$ \displaystyle du= \sin\theta d\theta $}
{$ \displaystyle \sin\theta d\theta $}{$ \displaystyle u=-\sin\theta $}{$ \displaystyle du= \cos\theta d\theta $}
{$ \displaystyle \sqrt{a^2 - x^2}dx $}{$ \displaystyle x= a\cos\theta $}{$ \displaystyle dx = -a\sin\theta d\theta $}
{$ \displaystyle \sqrt{a^2 - x^2}dx $}{$ \displaystyle x= a\sin\theta $}{$ \displaystyle dx = a\cos\theta d\theta $}
{$ \displaystyle \sqrt{a^2 + x^2}dx $}{$ \displaystyle x= a\tan\theta $}{$ \displaystyle dx= a\sec^2\theta d\theta $}
{$ \displaystyle \sqrt{a^2 + x^2} dx $}{$ \displaystyle x= a\sinh\theta $}{$ \displaystyle dx= a\cosh\theta d\theta $}
{$ \displaystyle \sqrt{x^2 - a^2}dx $}{$ \displaystyle x=a\sec\theta $}{$ \displaystyle dx= a\sec\theta\tan\theta d\theta $}

Trigonometric Manipulations

The following manipulations are useful when trigonometric function appear in the integrand, but they simply change the form of the integrand to an equivalent form. They do not change the limits of integrations or the differential.

These are not really a technique of integration, because they can be used with any trigonometric expression, anywhere. In other words, this is not calculus: it is remedial trigonometry.

Because they do not change the differential or the limits, these should be preferred if they make the integrand calculable.

Trigonometric Manipulations
IntegrandUse Identity
even power of {$ \displaystyle \sin\theta $}{$ \displaystyle \sin^2\theta = 1 - \cos^2\theta $}
even power of {$ \displaystyle \cos\theta $}{$ \displaystyle \cos^2\theta = {{1 + \cos 2\theta} \over 2} $}

Graph:

Sources:

  1. Prof. Miller, MIT, 18.01SC, Fall, 2010, Lecture 27
  2. Prof. Miller, MIT, 18.01SC, Fall, 2010, Lecture 28
  3. Joel Lewis, MIT, 18.01SC, Fall, 2010, Rescitation, Trig Integral Help
  4. Joel Lewis, MIT, 18.01SC, Fall, 2010, Rescitation, Hyperbolic Trig Substitution
  5. Prof. Leonard, Calculus 2 Lecture 7.3: Integrals By Trigonometric Substitution

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Category: Math


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August 06, 2017

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