In
mathematics, trigonometric substitution is the
replacement of
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
for other expressions. In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
to simplify certain
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
s containing
radical expressions. 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 ''a''2 − ''x''2
Let
, and use the
identity .
Examples of Case I
Example 1
In the integral
:
we may use
:
Then,
:
The above step requires that
and
. We can choose
to be the principal root of
, 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
goes from
to
, then
goes from
to
, so
goes from
to
. Then,
:
Some care is needed when picking the bounds. Because integration above requires that
,
can only go from
to
. Neglecting this restriction, one might have picked
to go from
to
, 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
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
.
Since
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 ''a''2 + ''x''2
Let
, and use the identity
.
Examples of Case II
Example 1
In the integral
:
we may write
:
so that the integral becomes
:
provided
.
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
so that
, and
by the range of arctangent, so that
and
.
Then,
:
The
integral of secant cubed
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
:\begin
\int \sec^3 x \, dx
&= \tfrac12\sec x \tan x + \tfrac12 \int \sec x\, dx + C \\ mu&= \tfrac12(\sec x \tan x + \ln \left, \sec x + \ta ...
may be evaluated using
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
. As a result,
:
Case III: Integrands containing ''x''2 − ''a''2
Let
, and use the identity
Examples of Case III
Integrals like
:
can also be evaluated by
partial fractions
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
rather than trigonometric substitutions. However, the integral
:
cannot. In this case, an appropriate substitution is:
:
where
so that
, and
by assuming
, 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
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
:\begin
\int \sec^3 x \, dx
&= \tfrac12\sec x \tan x + \tfrac12 \int \sec x\, dx + C \\ mu&= \tfrac12(\sec x \tan x + \ln \left, \sec x + \ta ...
by parts.
As a result,
:
When
, which happens when
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
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the ...
.
For example,
:
Hyperbolic substitution
Substitutions of
hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s can also be used to simplify integrals.
In the integral
, make the substitution
,
Then, using the identities
and
See also
*
Integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
*
Weierstrass substitution
*
Euler substitution
Euler substitution is a method for evaluating integrals of the form
\int R(x, \sqrt) \, dx,
where R is a rational function of x and \sqrt. In such cases, the integrand can be changed to a rational function by using the substitutions of Euler.
...
References
{{Integrals
Integral calculus
Trigonometry