The integral of secant cubed is a frequent and challenging
indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ...
of elementary
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 ...
:
:
where
is the inverse
Gudermannian function
In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwee ...
, the
integral of the secant function
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
:
\int \sec \theta ...
.
There are a number of reasons why this particular antiderivative is worthy of special attention:
* The technique used for reducing integrals of higher odd powers of secant to lower ones is fully present in this, the simplest case. The other cases are done in the same way.
* The utility of hyperbolic functions in integration can be demonstrated in cases of odd powers of secant (powers of tangent can also be included).
* This is one of several integrals usually done in a first-year calculus course in which the most natural way to proceed involves
integrating 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. ...
and returning to the same integral one started with (another is the integral of the product of an
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
with a sine or cosine function; yet another the integral of a power of the sine or cosine function).
* This integral is used in evaluating any integral of the form
::
: where
is a constant. In particular, it appears in the problems of:
:*
rectifying
A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction. The reverse operation (converting DC to AC) is performed by an inver ...
the
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
and the
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
:* finding the
surface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...
of the
helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarit ...
.
Derivations
Integration by parts
This
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
may be found by
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 follows:
:
where
:
Then
:
Next add
to both sides of the equality just derived:
:
given that the
integral of the secant function
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
:
\int \sec \theta ...
is
Finally, divide both sides by 2:
:
which was to be derived.
Reduction to an integral of a rational function
:
where
, so that
. This admits a decomposition 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 ...
:
:
Antidifferentiating term by-term, one gets
:
Hyperbolic functions
Integrals of the form:
can be reduced using the Pythagorean identity if
is even or
and
are both odd. If
is odd and
is even, hyperbolic substitutions can be used to replace the nested integration by parts with hyperbolic power reducing formulas.
:
Note that
follows directly from this substitution.
:
Higher odd powers of secant
Just as the integration by parts above reduced the integral of secant cubed to the integral of secant to the first power, so a similar process reduces the integral of higher odd powers of secant to lower ones. This is the secant reduction formula, which follows the syntax:
:
Alternatively:
:
Even powers of tangents can be accommodated by using binomial expansion to form an odd polynomial of secant and using these formulae on the largest term and combining like terms.
See also
*
Lists of integrals
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does no ...
Notes
References
{{Calculus topics
Integral calculus