This page covers the derivation and use of the secant reduction formula for integration.
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Secant Reduction Formula (n is an integer and \(n>1\)) |
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\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\) |
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 \((n-1)\). |
\(\displaystyle{ \int{\sec^nx~dx} = }\) \(\displaystyle{ \frac{\sec^{n-2}x\tan x}{n-1} }\) \(+\) \(\displaystyle{ \frac{n-2}{n-1}\int{\sec^{n-2}x~dx} }\) |
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This last equation is the secant reduction formula. Here is a video showing this same derivation.
video by PatrickJMT |
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Now let's work some practice problems.
Practice
Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.
\(\displaystyle{ \int{ \sec^3x ~dx } }\)
Problem Statement |
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Evaluate \(\displaystyle{ \int{ \sec^3x ~dx } }\) using integration by parts and then verify your answer using the secant reduction formula.
Solution |
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video by The Organic Chemistry Tutor |
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Log in to rate this practice problem and to see it's current rating. |
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Evaluate \(\displaystyle{ \int{ \sec^4x ~dx } }\) using the secant reduction formula.
Problem Statement |
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Evaluate \(\displaystyle{ \int{ \sec^4x ~dx } }\) using the secant reduction formula.
Solution |
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video by PatrickJMT |
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Log in to rate this practice problem and to see it's current rating. |
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Really UNDERSTAND Calculus
basic trig identities |
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\(\sin^2\theta+\cos^2\theta=1\) | \(1+\tan^2\theta=\sec^2\theta\) |
\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\) | \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\) |
\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\) | \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\) |
power reduction (half-angle) formulae |
\(\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}\) | \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\) |
double angle formulae |
\(\sin(2\theta)=2\sin\theta\cos\theta\) | \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\) |
links |
basic trig derivatives | ||
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\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = }\) \(\sec(t)\tan(t) \) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = }\) \( -\csc(t)\cot(t) \) | |
basic trig integrals | ||
\(\int{\sin(x)~dx} = -\cos(x)+C\) | ||
\(\int{\cos(x)~dx} = \sin(x)+C\) | ||
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) | ||
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | ||
\(\int{\sec(x)~dx} = \ln\abs{\sec(x)+\tan(x)}+C\) | ||
\(\int{\csc(x)~dx} = -\ln\abs{\csc(x)+\cot(x)}+C\) | ||
reduction formulae | ||
Reduction Formulas (n is a positive integer) | ||
\(\displaystyle{\int{\sin^n x~dx} = -\frac{\sin^{n-1}x\cos x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\sin^{n-2}x~dx} }\) | ||
\(\displaystyle{\int{\cos^n x~dx} = \frac{\cos^{n-1}x\sin x}{n} + }\) \(\displaystyle{ \frac{n-1}{n}\int{\cos^{n-2}x~dx}}\) | ||
Reduction Formulas (n is an integer and \(n>1\)) | ||
\(\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}\) | ||
\(\displaystyle{\int{\sec^n x~dx} = \frac{\sec^{n-2}x\tan x}{n-1} + }\) \(\displaystyle{ \frac{n-2}{n-1}\int{\sec^{n-2}x~dx}}\) | ||
links | ||
related topics on other pages |
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external links you may find helpful |
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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Practice Instructions
Unless otherwise instructed, evaluate these integrals directly, then check your answer using the reduction formula.