In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities
to simplify certain integrals containing the radical expressions
respectively.
To simplify
one would substitute a sin(θ) for x. This changes the integrand to
-
Factoring out of the radical gives
which becomes
Replacing 1 − sin2θ by cos2&theta, we get
Normally the square root of x2 becomes |x|, but if the integral is definite and cos(θ) is always non-negative on the interval [a,b], one does not have to write the "absolute value" and the integrand can be simplified to
In the expression a2 + x2, the substitution of a tan(θ) for x makes it possible to use the identity tan2θ + 1 = sec2θ.
Similarly, in x2 − a2, the substitution of sec(θ) for x makes it possible to use the identity sec2 − 1 = tan2.
Examples
In the integral
one may use
so that the integral becomes
(provided a > 0; if a < 0 then √a2 would be |a|, which would differ from a).
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 we have
In the integral
one may write
so that the integral becomes
(provided a > 0).
