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:[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


Integrals containing a2x2Edit

In the integral


we may use




The above step requires that a > 0 and cos(θ) > 0; we can choose a to be the positive square root of a2, and we impose the restriction π/2 < θ < π/2 on θ by using the arcsin function.

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


Some care is needed when picking the bounds. The integration above requires that π/2 < θ < π/2, so θ going from 0 to π/6 is the only choice. Neglecting this restriction, one might have picked θ to go from π to 5π/6, which would have resulted in the negative of the actual value.

Integrals containing a2 + x2Edit

In the integral


we may write


so that the integral becomes


provided a ≠ 0.

Integrals containing x2a2Edit

Integrals like


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


cannot. In this case, an appropriate substitution is:




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

Substitutions that eliminate trigonometric functionsEdit

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

For instance,


Hyperbolic substitutionEdit

Substitutions of hyperbolic functions can also be used to simplify integrals.[3]

In the integral  , make the substitution  ,  

Then, using the identities   and  


See alsoEdit


  1. ^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
  2. ^ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 0-321-58876-2.
  3. ^ Boyadzhiev, Khristo N. "Hyperbolic Substitutions for Integrals" (PDF). Retrieved 4 March 2013.