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. 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 a2 − x2

Let and use the identity

Examples of Case I

Example 1

In the integral we may use Then, The above step requires that a > 0 and We can choose a to be the principal root of a^2, 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 x goes from 0 to a/2, then \sin \theta goes from 0 to 1/2, so \theta goes from 0 to \pi / 6. Then, Some care is needed when picking the bounds. Because integration above requires that, \theta can only go from 0 to \pi / 6. Neglecting this restriction, one might have picked \theta to go from \pi to 5\pi /6, 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

The integral may be evaluated by letting where a > 0 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

Let and use the identity

Examples of Case II

Example 1

In the integral we may write so that the integral becomes provided a \neq 0. 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

The integral may be evaluated by letting where a > 0 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 x2 − a2

Let and use the identity

Examples of 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 a > 0 so that and by assuming x > 0, 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. As a result, When which happens when x < 0 given the range of arcsecant, meaning instead in that case.

Substitutions that eliminate trigonometric functions

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

Substitutions of hyperbolic functions can also be used to simplify integrals. For example, to integrate, introduce the substitution x=a\sinh{u} (and hence ), then use the identity to find: If desired, this result may be further transformed using other identities, such as using the relation :