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 and returning to the same integral one started with (another is the integral of the product of an
exponential function 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:
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:
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.
^Spivak, Michael (2008). "Integration in Elementary Terms". Calculus. p.
382. This is a tricky and important integral that often comes up.
^
abcStewart, James (2012). "Section 7.2: Trigonometric Integrals". Calculus - Early Transcendentals. United States: Cengage Learning. pp. 475–6.
ISBN978-0-538-49790-9.
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 and returning to the same integral one started with (another is the integral of the product of an
exponential function 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:
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:
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.
^Spivak, Michael (2008). "Integration in Elementary Terms". Calculus. p.
382. This is a tricky and important integral that often comes up.
^
abcStewart, James (2012). "Section 7.2: Trigonometric Integrals". Calculus - Early Transcendentals. United States: Cengage Learning. pp. 475–6.
ISBN978-0-538-49790-9.