You CAN Ace Calculus

### Topics You Need To Understand For This Page

 basic trig integration integration by substitution integration by parts

### 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

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### 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.

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

 Difference Between Trig Integration and Trig Substitution Trig Integration - Complete Case List Summary (One Angle) Integrands Involving Secant and/or Tangent Practice

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

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

Problem Statement

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

Solution

### 104 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 388 solution video

video by Krista King Math

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

Problem Statement

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

Solution

### 398 solution video

video by Krista King Math

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

Problem Statement

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

Solution

### 95 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 97 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 98 solution video

video by PatrickJMT

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

Problem Statement

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

Solution

### 91 solution video

video by Krista King Math

Intermediate Problems

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

Problem Statement

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

Solution