Trigonometric substitution

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions.  One may use the trigonometric identities to simplify certain integrals containing radical expressions:

Substitution 1. If the integrand contains a2 − x2, let
 
and use the identity
 

Substitution 2. If the integrand contains a2 + x2, let
 
and use the identity

Substitution 3. If the integrand contains x2 − a2, let

and use the identity

Examples

Integrals containing a2 − x2 

In the integral

we may use

Then,

The above step requires that  and ; we can choose  to be the positive square root of , and we impose the restriction  on  by using the arcsin function.

For a definite integral, one must figure out how the bounds of integration change.  For example, as  goes from 0 to , then  goes from 0 to 1/2, so  goes from 0 to .  Then,

Some care is needed when picking the bounds. The integration above requires that , so  going from 0 to π/6 is the only choice. Neglecting this restriction, one might have picked  to go from  to , which would have resulted in the negative of the actual value.

Integrals containing a2 + x2
In the integral

we may write

so that the integral becomes

provided .

Integrals containing x2 − a2
Integrals like

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

cannot.  In this case, an appropriate substitution is:

Then,

We can then solve this using the formula for the integral of secant cubed.

Substitutions that eliminate trigonometric functions
Substitution can be used to remove trigonometric functions.  In particular, see Tangent half-angle substitution.

For instance,

Hyperbolic substitution
Substitutions of hyperbolic functions can also be used to simplify integrals.

In the integral , make the substitution , 

Then, using the identities  and

See also

 Tangent half-angle substitution

References

Category:Integral calculus
Category:Trigonometry