Integral of secant cubed

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One of the more challenging indefinite integrals of elementary calculus is

$\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C.$

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.

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 appears in the problem of rectifying (i.e. finding the arc length of) the parabola.

This integral appears in the problem of finding the surface area of the helicoid.

[edit] Derivation

This antiderivative may be found by integration by parts, as follows:

$\int \sec^3 x \, dx = \int u\,dv$

where

$\begin{align} u &{}= \sec x, \\ dv &{}= \sec^2 x\,dx, \\ du &{}= \sec x \tan x\,dx, \\ v &{}= \tan x. \end{align}$

Then

$\begin{align} \int \sec^3 x \, dx &{}= \int u\,dv \\ &{}= uv - \int v\,du \\ &{} = \sec x \tan x - \int \sec x \tan^2 x\,dx \\ &{}= \sec x \tan x - \int \sec x\, (\sec^2 x - 1)\,dx \\ &{}= \sec x \tan x - \int \sec^3 x \, dx + \int \sec x\,dx. \end{align}$

Next we add $\scriptstyle{}\int\sec^3 x\,dx$ to both sides of the equality just derived:

$\begin{align} 2 \int \sec^3 x \, dx &{}= \sec x \tan x + \int \sec x\,dx \\ &{}= \sec x \tan x + \ln|\sec x + \tan x| + C. \end{align}$

Then divide both sides by 2:

$\int \sec^3 x \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C.$

[edit] 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:

$\int \sec^n x \, dx = \frac{\sec^{n-2} x \tan x}{n-1} \,+\, \frac{n-2}{n-1}\int \sec^{n-2} x \, dx \qquad \text{ (for }n \ne 1\text{)}\,\!$

Alternatively:

$\int \sec^n x \, dx = \frac{\sec^{n-1} x \sin x}{n-1} \,+\, \frac{n-2}{n-1}\int \sec^{n-2} x \, dx \qquad \text{ (for }n \ne 1\text{)}\,\!$

[edit] See also

Table of integrals

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