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AP Calculus AB: 9.5.1 An Overview of Trigonometric Substitution Strategy

Mathematics12 CardsCreated 8 months ago

This content introduces the strategy of trigonometric substitution for evaluating integrals involving square roots of sums or differences of squares. It outlines the process using right triangle relationships, highlights key substitution patterns, and explains how to simplify and revert integrals using trigonometric identities.

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An Overview of Trigonometric Substitution Strategy

• Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares.

  1. Match the square root expression with the sides of a right triangle.

  2. Substitute the corresponding trigonometric function into the integrand.

  3. Evaluate the resulting simpler integral.

  4. Convert from trigonometric functions back to the original variables.

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Key Terms

Term
Definition

An Overview of Trigonometric Substitution Strategy

• Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares.

  1. Match the s...

note

  • When you notice a radical expression or a rational power in an integrand, then the integral is a good candidate for trigonometric substitut...

Which of the following is the hypotenuse of the right triangle you might construct in order to evaluate the given integral?∫x^3/√−x^2+5x dx

5/2

For the following integral, which trig substitution should you use?∫b/2a 0 dx/[b^2−(ax)^2]^3/2 (a≠0,b≠0)

Let x=b/a sinθ.

For the following integral, which trig substitution should you use?∫ 2/3 0 x^3√9x^2+4 dx

Let x=2/3 tanθ

Which of these integrals is equivalent to the given integral?∫x^3√x^2−11 dx

11√11∫sec^4θdθ


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AP Calculus AB: 9.5.1 An Overview of Trigonometric Substitution Strategy
TermDefinition

An Overview of Trigonometric Substitution Strategy

• Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares.

  1. Match the square root expression with the sides of a right triangle.

  2. Substitute the corresponding trigonometric function into the integrand.

  3. Evaluate the resulting simpler integral.

  4. Convert from trigonometric functions back to the original variables.

note

  • When you notice a radical expression or a rational power in an integrand, then the integral is a good candidate for trigonometric substitution.

  • The fact that the square root of the difference must correspond to the leg of the right triangle follows from the Pythagorean theorem. The square root of the sum of the squares is equal to the length of the hypotenuse, so to get the radical expression to fit the theorem you will have to match the other leg with the negative term underneath the radical. The same reasoning applies to why the square root of the sum corresponds to the hypotenuse.

  • Once you have generated your triangle, you can create a whole list of potential substitutions just by writing down the trigonometric expressions and finding what they are equal to in the triangle.

  • The constant underneath the radical sign doesn’t have to be one. It can be any number. You can find the square root of any positive number in the real number system.

  • You don’t have to go back to the method of finding a trig substitution each time you evaluate one of these integrals. The table to the left illustrates exactly which substitutions you will find useful for each of the different possibilities.

  • Notice that there are three basic triangles you can create when you make a trig substitution. Each triangle has its own substitution that works best for it. Remember that you can switch the legs of a right triangle and not change it, so substitutions involving sine might be easier if you use cosine.

Which of the following is the hypotenuse of the right triangle you might construct in order to evaluate the given integral?∫x^3/√−x^2+5x dx

5/2

For the following integral, which trig substitution should you use?∫b/2a 0 dx/[b^2−(ax)^2]^3/2 (a≠0,b≠0)

Let x=b/a sinθ.

For the following integral, which trig substitution should you use?∫ 2/3 0 x^3√9x^2+4 dx

Let x=2/3 tanθ

Which of these integrals is equivalent to the given integral?∫x^3√x^2−11 dx

11√11∫sec^4θdθ


Which of these integrals is equivalent to the given integral?∫(4x+5−x^2)^3/2xdx

∫81cos4θ/3sinθ+2dθ

Which of these integrals is equivalent to the given integral?∫x^5√x^4+4 dx

2∫sin2θ/cos3θdθ

For the following integral, which trig substitution should you use?∫1 0 dx/√x^2−6x+5

Let x=3+2secθ

Which of these integrals is equivalent to the given integral?∫x/√3x+x^2dx

3/2∫(sec2θ−secθ)dθ

Which of these integrals is equivalent to the given integral?∫x^3√9x^4+6x^2−1 dx

1/18∫(√2sec2θ−secθ)dθ

Which of these integrals is equivalent to the given integral?∫√e^2x−9 dx

∫3tan2θdθ