Trigonometric substitution

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

edit

Let   and use the identity  

Examples of Case I

edit
 
Geometric construction for Case I

Example 1

edit

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

edit

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

edit

Let   and use the identity  

Examples of Case II

edit
 
Geometric construction for Case II

Example 1

edit

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

edit

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

edit

Let   and use the identity  

Examples of Case III

edit
 
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

edit

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

edit

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

edit

References

edit
  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.