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17Calculus Integrals - Secant Reduction Formula

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This page covers the derivation and use of the secant reduction formula for integration.

Secant Reduction Formula (n is an integer and \(n>1\))

\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\)

When you have an integral with only secant where the power is greater than one, you can use the secant reduction formula, repeatedly if necessary, to reduce the power until you end up with either \(\sec x\) or \(\sec^2 x\). Let's derive the formula and then work some practice problems.

Deriving The Secant Reduction Formula

Separate out a \(\sec^2x\) term.

\(\int{\sec^nx~dx} = \int{\sec^{n-2}x\sec^2x~dx}\)

Use integration by parts.

\(u=\sec^{n-2}x \to\) \(du=(n-2)(\sec^{n-3}x)[\sec x\tan x]~dx\)

\(dv = \sec^2x~dx \to v=\tan x\)

\(\int{\sec^nx~dx} =\) \(\sec^{n-2}x\tan x\) \(-\) \(\int{(n-2)\sec^{n-2}x\tan^2x~dx}\)

Use the identity \(\sec^2x = 1+\tan^2x\) to replace \(\tan^2x\) with \(\sec^2x-1\) in the last integral.

\(\int{\sec^nx~dx} =\) \(\sec^{n-2}x\tan x\) \(-\) \((n-2)\int{\sec^{n-2}x~(\sec^2x-1)~dx}\)

Distribute \((n-2)\sec^{n-2}x\) in the integral on the right and separate into two integrals.

\(\int{\sec^nx~dx} =\) \(\sec^{n-2}x\tan x\) \(-\) \((n-2)\int{\sec^nx~dx}\) \(+\) \((n-2)\int{\sec^{n-2}x~dx}\)

Add \((n-2)\int{\sec^nx~dx}\) to both sides of the equation, noticing that \(n-2+1 = n-1\).

\((n-1)\int{\sec^nx~dx} =\) \(\sec^{n-2}x\tan x\) \(+\) \((n-2)\int{\sec^{n-2}x~dx}\)

Solve for \(\int{\sec^nx~dx}\) by dividing both sides by \((n-1)\).

\(\displaystyle{ \int{\sec^nx~dx} = }\) \(\displaystyle{ \frac{\sec^{n-2}x\tan x}{n-1} }\) \(+\) \(\displaystyle{ \frac{n-2}{n-1}\int{\sec^{n-2}x~dx} }\)

This last equation is the secant reduction formula. Here is a video showing this same derivation.

PatrickJMT - \(\sec^n(x)\)

video by PatrickJMT

Now let's work some practice problems.

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Practice

Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.

\(\displaystyle{ \int{ \sec^3x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec^3x ~dx } }\) using integration by parts and then verify your answer using the secant reduction formula.

Solution

The Organic Chemistry Tutor - 2572 video solution

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\(\displaystyle{ \int{ \sec^4x ~dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \sec^4x ~dx } }\) using the secant reduction formula.

Solution

PatrickJMT - 96 video solution

video by PatrickJMT

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Practice Instructions

Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.

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