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AP Calculus AB: 9.5.2 Trigonometric Substitution Involving a Definite Integral: Part One

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This section explains how to apply trigonometric substitution to definite integrals involving square roots of sums or differences of squares. It emphasizes using triangle relationships and suggests treating the integral as indefinite first to avoid errors with limit conversion.

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Trigonometric Substitution Involving a Definite Integral: Part One

• 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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Definition

Trigonometric Substitution Involving a Definite Integral: Part One

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

  1. Match the s...

note

  • Trigonometric substitution can be used on definite integrals too. The same circumstances should be in place to use trigonometric substituti...

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AP Calculus AB: 9.5.2 Trigonometric Substitution Involving a Definite Integral: Part One
TermDefinition

Trigonometric Substitution Involving a Definite Integral: Part One

• 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

  • Trigonometric substitution can be used on definite integrals too. The same circumstances should be in place to use trigonometric substitution on a definite integral. There should be a radical made up of the difference or sum of squares, and the integral should not lend itself to a simple u-substitution.

  • There are two ways to work a definite integral. One way is to convert the limits of integration when you make your trigonometric substitution. It is recommended that you avoid that process, however, since there are lots of places where you might make a mistake. The better option is to ignore the limits of integration and work the problem like an indefinite integral first.

  • Here, the integral has a radical made up of the difference of squares. So set it equal to one of the legs of the triangle. Since the x-term is positive, it must represent the hypotenuse.

  • Notice that when you take the derivative of the cosine term to find dx you get a result with x-terms in it. Don’t let that confuse you. Just substitute out those x-terms with another trigonometric substitution.