Integral and Trigonometric Relationships

Here are the top 30 Integral and Trigonometric Identities, along with the latex syntax and any explanatory text for each identity:
Trigonometric RelationshipsSum-to-Product Formulas
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sin⁡(x)+sin⁡(y)=2sin⁡(x+y2)cos⁡(x−y2) cos(x)+cos⁡(y)=2cos⁡(x+y2)cos⁡(x−y2) sin(x)−sin⁡(y)=2sin⁡(x−y2)cos⁡(x+y2) cos(x)−cos⁡(y)=−2sin⁡(x+y2)sin⁡(x−y2)\sin(x)+\sin(y) = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\\ \\ cos(x)+\cos(y) = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\\ \\sin(x)-\sin(y) = 2\sin\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right) \\\ \\cos(x)-\cos(y) = -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)sin(x)+sin(y)=2sin(2x+y​)cos(2x−y​) cos(x)+cos(y)=2cos(2x+y​)cos(2x−y​) sin(x)−sin(y)=2sin(2x−y​)cos(2x+y​) cos(x)−cos(y)=−2sin(2x+y​)sin(2x−y​)
📝 Note
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Difference-to-Product Formulas
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sin⁡(x)cos⁡(y)=12(sin⁡(x+y)+sin⁡(x−y)) cos(x)sin⁡(y)=12(sin⁡(x+y)−sin⁡(x−y)) cos(x)cos⁡(y)=12(cos⁡(x+y)+cos⁡(x−y)) sin(x)sin⁡(y)=12(cos⁡(x+y)−cos⁡(x−y))\sin(x)\cos(y) = \frac{1}{2}\left(\sin(x+y)+\sin(x-y)\right) \\\ \\cos(x)\sin(y) = \frac{1}{2}\left(\sin(x+y)-\sin(x-y)\right) \\\ \\cos(x)\cos(y) = \frac{1}{2}\left(\cos(x+y)+\cos(x-y)\right) \\\ \\sin(x)\sin(y) = \frac{1}{2}\left(\cos(x+y)-\cos(x-y)\right)sin(x)cos(y)=21​(sin(x+y)+sin(x−y)) cos(x)sin(y)=21​(sin(x+y)−sin(x−y)) cos(x)cos(y)=21​(cos(x+y)+cos(x−y)) sin(x)sin(y)=21​(cos(x+y)−cos(x−y))
📝 Note
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Double Angle Formulas
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sin⁡2θ=2sin⁡(θ)cos⁡(θ)cos⁡2θ=cos⁡2θ−sin⁡2θtan⁡2θ=2tan⁡θ1−tan⁡2θ\sin 2\theta = 2\sin (\theta)\cos(\theta) \\ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \\ \tan 2\theta = \frac {2\tan\theta} {1-\tan^2 \theta}sin2θ=2sin(θ)cos(θ)cos2θ=cos2θ−sin2θtan2θ=1−tan2θ2tanθ​
Pythagorean Identities﻿
sin⁡2(x)+cos⁡2(x)=1sin⁡2(x)=1−cos⁡2(x)cos⁡2(x)=1−sin⁡2(x) 1+tan⁡2(x)=sec⁡2(x)1+cot⁡2(x)=csc⁡2(x)\sin^2(x) + \cos^2(x) = 1 \\ \sin^2(x) = 1-\cos^2(x) \\ \cos^2(x) = 1- \sin^2(x) \\\ \\ 1+ \tan^2 (x) = \sec ^2 (x) \\ 1 + \cot ^2 (x) = \csc ^2 (x) sin2(x)+cos2(x)=1sin2(x)=1−cos2(x)cos2(x)=1−sin2(x) 1+tan2(x)=sec2(x)1+cot2(x)=csc2(x)
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Cofunction Identity
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sin⁡(π/2−x)=cos⁡(x) cos(π/2−x)=sin⁡(x)\sin(\pi/2 - x) = \cos(x) \\\ \\cos(\pi/2 - x) = \sin(x)sin(π/2−x)=cos(x) cos(π/2−x)=sin(x)
These identities are used to convert between sine and cosine in situations where one is more convenient to use.
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Reciprocal Identity
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sec⁡(x)=1sin⁡(x) csc⁡(x)=1cos⁡(x)\sec (x) = \frac{1}{\sin(x)} \\\ \\ \csc(x)  = \frac{1}{\cos(x)}sec(x)=sin(x)1​ csc(x)=cos(x)1​
These identities are used to simplify expressions and to find the inverse trigonometric functions.
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Quotient Identity
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tan⁡(x)=sin⁡(x)cos⁡(x) cot(x)=cos⁡(x)sin⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)} \\\ \\cot(x) = \frac{\cos(x)}{\sin(x)}tan(x)=cos(x)sin(x)​ cot(x)=sin(x)cos(x)​
📝 Note
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Fundamental Integrals
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∫dx=x+C ∫exdx=ex+C ∫xndx=xn+1n+1+C ∫axdx=axln⁡(a)+C ∫eaxdx=eaxa+C ∫1x=ln⁡∣x∣+C\int dx = x+C  \\\ \\ \int e^x dx = e^x + C \\\ \\ \int x^n dx = \frac {x^{n+1}}{n+1} + C  \\\ \\ \int a^x dx = \frac{a^x}{\ln(a)} + C\\\ \\ \int e^{ax} dx = \frac {e^{ax}} {a} + C \\\ \\ \int \frac{1}{x} = \ln |x| + C  ∫dx=x+C ∫exdx=ex+C ∫xndx=n+1xn+1​+C ∫axdx=ln(a)ax​+C ∫eaxdx=aeax​+C ∫x1​=ln∣x∣+C
Trigonometric Integral relationships﻿
∫sin⁡(x) dx=−cos⁡(x)+C∫cos⁡(x) dx=sin⁡(x)+C∫tan⁡(x) dx=ln⁡∣sec⁡(x)∣+C=−ln⁡∣cos(x)∣+C ∫sec⁡(x) dx=ln⁡∣(sec⁡x+tan⁡x)∣+C∫csc⁡(x)dx=−ln⁡∣csc⁡(x)+cot⁡(x)∣+C∫cot⁡(x)dx=ln⁡∣sin⁡(x)∣+C∫csc⁡(x)cot⁡(x) dx=−csc⁡(x)+C ∫sec⁡2(x) dx=tan⁡(x)+C∫csc⁡2(x) dx=−cot⁡(x)+C∫cot2(x)dx=−cot⁡(x)−x+C dx ∫11−x2dx=sin⁡−1(x)+C ∫1a2−x2dx=1asin⁡−1(xa)+C∫1a2+x2dx=tan⁡−1(xa)+C\int\sin(x) \: dx = -\cos(x) + C \\ \int \cos (x) \: dx = \sin (x)  +  C \\ \int \tan (x) \: dx  = \ln |\sec (x)| + C = -\ln|cos(x)| + C \\\ \\ \int\sec(x) \:dx = \ln |(\sec x + \tan x )| + C  \\ \int \csc(x) dx =   -\ln | \csc (x) + \cot (x) | + C \\ \int \cot (x) dx = \ln | \sin (x) | + C \\  \int \csc(x) \cot(x)  \: dx = -\csc (x) + C\\\ \\ \int\sec^2(x) \: dx = \tan(x) + C \\ \int\csc^2(x) \: dx = -\cot(x) + C \\  \int cot^2(x) dx = -\cot(x) -x + C  \: dx   \\\ \\  \int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1}(x) + C \\\  \int\frac{1}{\sqrt{a^2-x^2}}dx = \frac{1}{a}\sin^{-1}(\frac{x}{a}) + C \\ \int\frac{1}{\sqrt{a^2 + x^2}}dx = \tan^{-1}(\frac{x}{a}) + C∫sin(x)dx=−cos(x)+C∫cos(x)dx=sin(x)+C∫tan(x)dx=ln∣sec(x)∣+C=−ln∣cos(x)∣+C ∫sec(x)dx=ln∣(secx+tanx)∣+C∫csc(x)dx=−ln∣csc(x)+cot(x)∣+C∫cot(x)dx=ln∣sin(x)∣+C∫csc(x)cot(x)dx=−csc(x)+C ∫sec2(x)dx=tan(x)+C∫csc2(x)dx=−cot(x)+C∫cot2(x)dx=−cot(x)−x+Cdx ∫1−x2​1​dx=sin−1(x)+C ∫a2−x2​1​dx=a1​sin−1(ax​)+C∫a2+x2​1​dx=tan−1(ax​)+C
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Trigonometric Integral relationships
Even More Relationships!Constant Multiple Rule:
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∫kf (x) dx=k∫f(x) dx\int kf\: (x) \: dx = k \int f(x) \: dx \\∫kf(x)dx=k∫f(x)dx
📝 Note! 
K is a constant
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Sum Rule
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∫(f(x)+g(x))dx=∫f(x)dx+∫g(x)dx\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx∫(f(x)+g(x))dx=∫f(x)dx+∫g(x)dx
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Difference Rule
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∫(f(x)−g(x))dx=∫f(x)dx−∫g(x)dx\int (f(x) - g(x)) dx = \int f(x) dx - \int g(x) dx∫(f(x)−g(x))dx=∫f(x)dx−∫g(x)dx
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Power Rule
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∫xndx=xn+1n+1+C\int x^n dx = \frac{x^{n+1}}{n+1} + C∫xndx=n+1xn+1​+C
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Logarithmic Rule
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∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C∫x1​dx=ln∣x∣+C
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Exponential Rule
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∫exdx=ex+C\int e^x dx = e^x + C∫exdx=ex+C
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Integral of constant raised to a variable
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∫axdx=axln(a)+C\int a^x dx = \frac{a^x}{ln(a)} + C∫axdx=ln(a)ax​+C
📝 Note!
a>0, a!=1
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Other Hyperbolic Trigonometric Relationships﻿
sinh⁡x=ex−e−x2cosh⁡x=ex+e−x2tanh⁡x=sinh⁡xcosh⁡xcoth⁡x=cosh⁡xsinh⁡xcsch x=1sinh⁡x\sinh x = \frac{e^x - e^{-x}}{2}\\\cosh x = \frac{e^x + e^{-x}}{2} \\ \tanh x = \frac{\sinh x}{\cosh x} \\\coth x = \frac{\cosh x}{\sinh x} \\ \text{csch}\:  x = \frac{1}{\sinh x}sinhx=2ex−e−x​coshx=2ex+e−x​tanhx=coshxsinhx​cothx=sinhxcoshx​cschx=sinhx1​
Even more integrals:Integration by Parts 
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∫udv=uv−∫vdu\int udv = uv - \int vdu∫udv=uv−∫vdu
📝 Note!
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For Trigonometric Substitution:
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Let x=a sin(θ),or x=a tan(θ),or x=a sec(θ)Let \ x = a \: sin(θ), \\ or \ x = a \: tan(θ), \\ or \ x = a \: sec(θ)Let x=asin(θ),or x=atan(θ),or x=asec(θ)
Then use trigonometric identities to integrate.
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Improper Integral:
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∫a∞f(x)dx=lim⁡b→∞∫abf(x)dx\int_a^{\infty} f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx∫a∞​f(x)dx=limb→∞​∫ab​f(x)dx
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Beta Function:
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∫01xm−1(1−x)n−1dx=B(m,n)\int_0^1 x^{m-1} (1-x)^{n-1} dx = B(m,n)∫01​xm−1(1−x)n−1dx=B(m,n)
📝 Note
In physics, the beta function is used to compute and represent the scatter amplitude of a quantum hadro-dynamic concept known as Regge trajectory
Used in calculus applications
The beta function can also be used to evaluate the cumulative distribution function of a beta distribution.
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Gamma Function:
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∫0∞xm−1e−xdx=Γ(m)\int_0^{\infty} x^{m-1} e^{-x} dx = \Gamma(m)∫0∞​xm−1e−xdx=Γ(m)
📝 Note
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Elliptic Integral:
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∫0aa2−x2dx=a22sin⁡−1(xa)+x2a2−x2\int_{0}^{a} \sqrt{a^2 - x^2} dx = \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + \frac{x}{2} \sqrt{a^2 - x^2}∫0a​a2−x2​dx=2a2​sin−1(ax​)+2x​a2−x2​
📝 Note
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