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 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

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

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 $$n>1$$)

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

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

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

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

Cosine Reduction Formula (where n is a positive integer)

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

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

Deriving The Cosine Reduction Formula

$$\displaystyle{ \int{\cos^nx~dx} = \frac{\cos^{n-1}x\sin x}{n} + \frac{n-1}{n}\int{\cos^{n-2}x~dx} }$$ Separate out one $$\cos x$$ term. $$\int{\cos^nx~dx} = \int{\cos^{n-1}x \cos x~dx}$$ Use integration by parts. $$u=\cos^{n-1}x \to$$ $$du=(n-1)\cos^{n-2}x(-\sin x)~dx$$ $$dv=\cos x~dx \to v=\sin x$$ $$\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$\int{(n-1)\cos^{n-2}x\sin^2x~dx}$$ Use $$\sin^2x + \cos^2x = 1$$ to replace the $$\sin^2x$$ in the last integral with $$1-\cos^2x$$. $$\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$(n-1)\int{\cos^{n-2}x~(1-\cos^2x)~dx}$$ In the last integral, distribute the $$\cos^{n-2}x$$ term and separate the integral into two integrals. Don't forget to distribute the $$(n-1)$$ term as well. $$\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$(n-1)\int{\cos^{n-2}x~dx}$$ $$-$$ $$(n-1)\int{\cos^nx~dx}$$ Now add $$(n-1)\int{\cos^nx~dx}$$ to both sides giving us $$n-1+1=n$$ on the left. $$n\int{\cos^nx~dx} =$$ $$\sin x\cos^{n-1}x$$ $$+$$ $$(n-1)\int{\cos^{n-2}x~dx}$$ Solve for $$\int{\cos^nx~dx}$$ by dividing both sides by n.

This last equation is the cosine reduction formula. Now let's work some practice problems.