Trigonometric substitution

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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[edit]

Integrals containing a2x2[edit]

In the integral

we may use

Then,

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 + x2[edit]

In the integral

we may write

so that the integral becomes

provided a ≠ 0.

Integrals containing x2a2[edit]

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[edit]

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

For instance,

Hyperbolic substitution[edit]

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[edit]

References[edit]

  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.