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Trigonometric substitution

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In mathematics, a trigonometric substitution replaces a trigonometric function for another expression. In calculus, trigonometric substitutions are a technique for evaluating integrals. In this case, an expression involving a radical function is replaced with a trigonometric one. Trigonometric identities may help simplify the answer.[1][2] Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.

Case I: Integrands containing a2x2

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Let and use the identity

Examples of Case I

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Geometric construction for Case I

Example 1

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In the integral

we may use

Then,

The above step requires that and We can choose to be the principal root of and impose the restriction by using the inverse sine function.

For a definite integral, one must figure out how the bounds of integration change. For example, as goes from to then goes from to so goes from to Then,

Some care is needed when picking the bounds. Because integration above requires that , can only go from to Neglecting this restriction, one might have picked to go from to which would have resulted in the negative of the actual value.

Alternatively, fully evaluate the indefinite integrals before applying the boundary conditions. In that case, the antiderivative gives

as before.

Example 2

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The integral

may be evaluated by letting where so that and by the range of arcsine, so that and

Then,

For a definite integral, the bounds change once the substitution is performed and are determined using the equation with values in the range Alternatively, apply the boundary terms directly to the formula for the antiderivative.

For example, the definite integral

may be evaluated by substituting with the bounds determined using

Because and

On the other hand, direct application of the boundary terms to the previously obtained formula for the antiderivative yields as before.

Case II: Integrands containing a2 + x2

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Let and use the identity

Examples of Case II

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Geometric construction for Case II

Example 1

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In the integral

we may write

so that the integral becomes

provided

For a definite integral, the bounds change once the substitution is performed and are determined using the equation with values in the range Alternatively, apply the boundary terms directly to the formula for the antiderivative.

For example, the definite integral

may be evaluated by substituting with the bounds determined using

Since and

Meanwhile, direct application of the boundary terms to the formula for the antiderivative yields same as before.

Example 2

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The integral

may be evaluated by letting

where so that and by the range of arctangent, so that and

Then, The integral of secant cubed may be evaluated using integration by parts. As a result,

Case III: Integrands containing x2a2

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Let and use the identity

Examples of Case III

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Geometric construction for Case III

Integrals such as

can also be evaluated by partial fractions rather than trigonometric substitutions. However, the integral

cannot. In this case, an appropriate substitution is:

where so that and by assuming so that and

Then,

One may evaluate the integral of the secant function by multiplying the numerator and denominator by and the integral of secant cubed by parts.[3] As a result,

When which happens when given the range of arcsecant, meaning instead in that case.

Substitutions that eliminate trigonometric functions

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Substitution can be used to remove trigonometric functions.

For instance,

The last substitution is known as the Weierstrass substitution, which makes use of tangent half-angle formulas.

For example,

Hyperbolic substitution

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Substitutions of hyperbolic functions can also be used to simplify integrals.[4]

For example, to integrate , introduce the substitution (and hence ), then use the identity to find:

If desired, this result may be further transformed using other identities, such as using the relation :

See also

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References

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  1. ^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.
  2. ^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
  3. ^ Stewart, James (2012). "Section 7.2: Trigonometric Integrals". Calculus - Early Transcendentals. United States: Cengage Learning. pp. 475–6. ISBN 978-0-538-49790-9.
  4. ^ Boyadzhiev, Khristo N. "Hyperbolic Substitutions for Integrals" (PDF). Archived from the original (PDF) on 26 February 2020. Retrieved 4 March 2013.