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You CAN Ace Calculus

Topics You Need To Understand For This Page

Trig Identities and Formulas

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

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

links

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

Trig Derivatives and Integrals

basic trig derivatives

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

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

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

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

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

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

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

17calculus trig formulas - full list

Related Topics and Links

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

learning and study techniques

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.]

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.

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 on the secant reduction formula page.

 

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 or use the reduction formula found on the tangent reduction formula page

 

Case 3 - even secant term: \( \int{ \sec^m(x)~\tan^n(x) ~dx} \) with even m

Factor out \(\sec^2(x)\), use \(\sec^2x = 1 + \tan^2x\) and let \(u = \tan(x)\).

 

Case 4 - odd tangent term: \( \int{ \sec^m(x)~\tan^n(x) ~dx} \) with odd n

Factor out \(\sec(x)\tan(x)\), use \(\sec^2x = 1 + \tan^2x\) and let \(u = \sec(x)\).

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Secant/Tangent Integration - Practice Problems Conversion

[A01-103] - [A02-104] - [A03-388] - [A04-298] - [B01-116] - [B02-121]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

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

Basic Problems

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

Problem Statement

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

Solution

103 solution video

video by PatrickJMT

close solution

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

Problem Statement

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

Solution

104 solution video

video by PatrickJMT

close solution

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

Problem Statement

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

Solution

388 solution video

video by Krista King Math

close solution

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

Problem Statement

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

Solution

398 solution video

video by Krista King Math

close solution

\(\displaystyle{ \int{ \sec^4x ~\tan^2x ~dx } }\)

Problem Statement

\(\displaystyle{ \int{ \sec^4x ~\tan^2x ~dx } }\)

Solution

95 solution video

video by PatrickJMT

close solution

\(\displaystyle{ \int{ \sec^3x ~\tan^5x ~dx } }\)

Problem Statement

\(\displaystyle{ \int{ \sec^3x ~\tan^5x ~dx } }\)

Solution

97 solution video

video by PatrickJMT

close solution

\(\displaystyle{ \int{ \tan(x) ~dx } }\)

Problem Statement

\(\displaystyle{ \int{ \tan(x) ~dx } }\)

Solution

98 solution video

video by PatrickJMT

close solution

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

Problem Statement

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

Solution

91 solution video

video by Krista King Math

close solution

Intermediate Problems

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

Problem Statement

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

Solution

121 solution video

close solution
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