1. Examples
     Integrals containing a² − x²   Integrals containing a² + x²  Integrals containing x² − a² 

2. Substitutions that eliminate trigonometric functions

3. Hyperbolic substitution

4. See also

5. References

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:^[1][2]

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 {{math|a > 0}} and {{math|cos(θ) > 0}}; we can choose {{mvar|a}} to be the positive square root of {{math|a²}}, and we impose the restriction {{math|−π/2 < θ < π/2}} on {{mvar|θ}} by using the arcsin function.

For a definite integral, one must figure out how the bounds of integration change. For example, as {{mvar|x}} goes from 0 to {{math|a/2}}, then {{math|sin θ}} goes from 0 to 1/2, so {{mvar|θ}} goes from 0 to {{math|π/6}}. Then,

Some care is needed when picking the bounds. The integration above requires that {{math|−π/2 < θ < π/2}}, so {{mvar|θ}} going from 0 to π/6 is the only choice. Neglecting this restriction, one might have picked {{mvar|θ}} to go from {{mvar|π}} to {{math|5π/6}}, 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 {{math|a ≠ 0}}.

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.^[3]

In the integral , make the substitution ,

Then, using the identities and

See also

• Tangent half-angle substitution

References

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