Trigonometric substitution

Summary

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions.[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 a2 − x2

Let , and use the identity .

Examples of Case I

Geometric construction for Case I

Example 1

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

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 .

Since 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

Let , and use the identity .

Examples of Case II

Geometric construction for Case II

Example 1

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

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 x2 − a2

Let , and use the identity

Examples of Case III

Geometric construction for Case III

Integrals like

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

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

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

In the integral , make the substitution ,

Then, using the identities and

See also

References

  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). Retrieved 4 March 2013.