Here are the top 30 Integral and Trigonometric Identities, along with the latex syntax and any explanatory text for each identity:

## Trigonometric Relationships

1. Sum-to-Product Formulas
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### 📝 Note

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1. Difference-to-Product Formulas
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### 📝 Note

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1. Double Angle Identities
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## Pythagorean Identities

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1. Cofunction Identity
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These identities are used to convert between sine and cosine in situations where one is more convenient to use.
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1. Reciprocal Identity
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These identities are used to simplify expressions and to find the inverse trigonometric functions.
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1. Quotient Identity
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### 📝 Note

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1. Fundamental Integrals
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## Even More Relationships!

1. Constant Multiple Rule:
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📝 Note!
K is a constant
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1. Sum Rule
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1. Difference Rule
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1. Power Rule
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1. Logarithmic Rule
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1. Exponential Rule
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1. Integral of constant raised to a variable
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📝 Note!
a>0, a!=1
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## Even more integrals:

1. Integration by Parts
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📝 Note!
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1. For Trigonometric Substitution:
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Then use trigonometric identities to integrate.
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1. Improper Integral:
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1. Beta Function:
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### 📝 Note

1. In physics, the beta function is used to compute and represent the scatter amplitude of a quantum hadro-dynamic concept known as Regge trajectory
2. Used in calculus applications
1. The beta function can also be used to evaluate the cumulative distribution function of a beta distribution.
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1. Gamma Function:
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### 📝 Note

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1. Elliptic Integral:
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