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You CAN Ace Calculus

17calculus > integrals > trig integration > secant-tangent integration

Trig Identities and Formulae - NEW

basic trig identities

$$\sin^2\theta+\cos^2\theta=1$$   |   $$1+\tan^2\theta=\sec^2\theta$$

$$\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}$$   |   $$\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}$$

$$\displaystyle{\sec\theta=\frac{1}{\cos\theta}}$$   |   $$\displaystyle{\csc\theta=\frac{1}{\sin\theta}}$$

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

$$\sin(2\theta)=2\sin\theta\cos\theta$$   |   $$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Trig Derivatives and Integrals - NEW

basic trig derivatives

$$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$

$$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$

$$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$

$$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$

$$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$

$$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

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 (where n is a positive integer)

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

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

Reduction Formulas (where n is an integer and $$n>1$$)

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

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

17calculus trig formulas - full list

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Secant-Tangent Trig Integration

Difference Between Trig Integration and Trig Substitution

Trig integration, covered on this page, is the evaluation of integrals that already have trig functions in the integrand.

Trig substitution is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. Once the integral is completely transformed, then trig integration is used to evaluate the integral. Once the evaluation is complete, another set of substitutions, based on the original ones, is done to convert the result back to the original variable.

This page covers integration of functions involving secants and/or tangents in more advanced form that require techniques other than just integration by substitution. [If you are first learning secant and tangent in integration, check out the basics of trig integration page.]

Trig Integration - Complete Case List Summary (One Angle)

Trig Integration - Complete Case List Summary (One Angle)

$$\int{\sin^mx \cos^nx~dx}$$

$$n=0$$

$$\int{\sin^mx~dx}$$

use reduction formula

$$m=0$$

$$\int{\cos^mx~dx}$$

use reduction formula

odd m

$$\int{\sin^{2k+1}x\cos^nx~dx}$$

factor out $$\sin x$$, use $$\sin^2x=1-\cos^2x$$ and let $$u=\cos x$$

odd n

$$\int{\sin^mx\cos^{2k+1}x~dx}$$

factor out $$\cos x$$, use $$\cos^2x=1-\sin^2x$$ and let $$u=\sin x$$

even m and n

$$\int{\sin^{2k}x\cos^{2p}x~dx}$$

use half-angle formulas

$$\int{\sec^mx \tan^nx~dx}$$

$$n=0$$

$$\int{\sec^mx~dx}$$

use reduction formula

$$m=0$$

$$\int{\tan^nx~dx}$$

use reduction formula or use $$\sec^2x=1+\tan^2x$$, expand out
and try one of the following two cases

even m

$$\int{\sec^{2k}x\tan^nx~dx}$$

factor out $$\sec^2x$$, use $$\sec^2x=1+\tan^2x$$ and let $$u=\tan x$$

odd n

$$\int{\sec^mx\tan^{2k+1}x~dx}$$

factor out $$\sec x\tan x$$, use $$\sec^2x=1+\tan^2x$$ and let $$u=\sec x$$

none of the above 4 cases hold

convert trig functions to sine and cosine and
try the sine/cosine techniques

Integrands Involving Secant and/or Tangent

In order to choose the technique you need to use, you need to determine the form of the integrand.

 Case 1 ( one secant term only ): $$\int{ \sec^n(x) ~dx}$$ Use the reduction formula found in the section below. Case 2 ( one tangent term only ): $$\int{ \tan^n(x) ~dx}$$ Use $$\sec^2x = 1 + \tan^2x$$, expand out and see if you can use case 3 and/or case 4 on the terms. Case 3: $$\int{ \sec^m(x)~\tan^n(x) ~dx}$$ with even m Factor out $$\sec^2(x)$$, use $$\sec^2x = 1 + \tan^2x$$ and let $$u = \tan(x)$$. Case 4: $$\int{ \sec^m(x)~\tan^n(x) ~dx}$$ with odd n Factor out $$\sec(x)\tan(x)$$, use $$\sec^2x = 1 + \tan^2x$$ and let $$u = \sec(x)$$.

Reduction Formulas

Having reduction formulas can save time when evaluating some integrals. These videos derive the reduction formula for a couple of common integrands.

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

In this next video, he uses the reduction formula for $$\tan^n(x)$$ to calculate an integral involving $$\tan^5(x)$$.

 Dr Chris Tisdell - $$\tan^n(x)$$

Reduction Formulas (where n is an integer and $$n>1$$)

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

derivation

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

derivation

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Practice Problems

Instructions - - Unless otherwise instructed, evaluate the following integrals using the techniques on this page. Give all answers in exact, simplified form.

 Here are a few practice problems. You can find more practice problems on this page.
 Level A - Basic

Practice A01

$$\displaystyle{\int{\sec^4x~\tan^6x~dx}}$$

solution

Practice A02

$$\displaystyle{\int{\frac{1}{\cos^6(x)\cot^2(x)}~dx}}$$

solution

Practice A03

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

solution

Practice A04

$$\displaystyle{\int{\tan^4x~\sec^6x~dx}}$$

solution

 Level B - Intermediate

Practice B01

$$\displaystyle{\int{\tan^5x~dx}}$$

solution

Practice B02

$$\displaystyle{\int{\sec(x)~dx}}$$

solution

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