17Calculus Integrals - Tangent Reduction Formula

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

Tangent Reduction Formula (n is an integer and $$n>1$$)

$$\displaystyle{\int{\tan^n x~dx}= \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx}}$$

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

Deriving The Tangent Reduction Formula

$$\displaystyle{ \int{\tan^nx~dx} = \frac{\tan^{n-1}x}{n-1} - \int{\tan^{n-2}x~dx} }$$ Separate out a $$\tan^2x$$. $$\int{\tan^nx~dx} = \int{\tan^{n-2}x \tan^2x~dx}$$ Use the identity $$\sec^2x=1+\tan^2x$$ to replace the $$\tan^2x$$ with $$\sec^2x-1$$. $$\int{\tan^nx~dx} = \int{\tan^{n-2}x~(\sec^2x-1)~dx}$$ Distribute the $$\tan^{n-2}x$$ term and separate into two integrals. $$\int{\tan^nx~dx} =$$ $$\int{\tan^{n-2}x \sec^2x~dx}$$ $$-$$ $$\int{\tan^{n-2}x~dx}$$ Use integration by substitution on the first integral on the right side of the equal sign. $$u=\tan x \to du=\sec^2x~dx$$ $$\int{\tan^nx~dx} =$$ $$\int{u^{n-2}~du}$$ $$-$$ $$\int{\tan^{n-2}x~dx}$$ Integrate the term containing u and convert back to x's. $$\displaystyle{ \int{\tan^nx~dx} = }$$ $$\displaystyle{ \frac{u^{n-1}}{n-1} - \int{\tan^{n-2}x~dx} }$$ This last equation is the tangent reduction formula.

Now let's work some practice problems.

Practice

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

$$\displaystyle{ \int{ \tan^3x ~dx } }$$

Problem Statement

Evaluate $$\displaystyle{ \int{ \tan^3x ~dx } }$$ using the techniques on the tangent-secant trig integration page. Then check your answer using the tangent reduction formula.

Solution

The Organic Chemistry Tutor - 2574 video solution

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$$\displaystyle{ \int{ \tan^4x ~dx } }$$

Problem Statement

Evaluate $$\displaystyle{ \int{ \tan^4x ~dx } }$$ using the techniques on the tangent-secant trig integration page. Then check your answer using the tangent reduction formula.

Solution

The Organic Chemistry Tutor - 2573 video solution

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Evaluate $$\int{ \tan^5x ~dx }$$ using the tangent reduction formula.

Problem Statement

Evaluate $$\int{ \tan^5x ~dx }$$ using the tangent reduction formula.

Solution

Here are two videos, by two different instructors, solving this problem.

PatrickJMT - 116 video solution

video by PatrickJMT

The Organic Chemistry Tutor - 116 video solution

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