Formulas

Basic Integration Rules

• (for )
• (for )
• (for )

Exponential Integrals

• (for )
• (for )

Trigonometric Integrals

Hyperbolic Integrals

Inverse Trigonometric Integrals

• (for )
• (for )
• (for )
• (for )
• (for )

Integration Techniques

Integration by Parts

Substitution Rule for Definite Integrals

Hermite-Ostrogradski Formula

Fundamental Theorem of Calculus

• Part 1:
• Part 2: where
• Leibniz Rule:

Properties of Definite Integrals

• Linearity:
• Additivity:
• Reversal:
• Zero-width:
• Even Function: (if )
• Odd Function: (if )
• Average Value:

Applications of Definite Integrals

• Area Between Curves: (where )
• Volume by Disk Method:
• Volume by Washer Method:
• Volume by Shell Method: (revolving about -axis)
• Arc Length:
• Parametric Arc Length:
• Surface Area of Revolution: (about -axis)

Improper Integrals

• Type 1 (Infinite Interval):
• Type 2 (Discontinuity at ):
• -Integral Test: converges if , diverges if

Trigonometric Identities

• Power Reduction: ,
• Product-to-Sum:
• Pythagorean: , ,
• Universal Substitution: gives:

Hyperbolic Identities

• Power Reduction: ,
• Pythagorean: ,
• Definitions: , ,

Reduction Formulas

• Sine Powers:
• Cosine Powers:
• Power-Exponential:

Functions of Several Variables

Partial Derivatives

• Partial derivative w.r.t. :
• Chain Rule (1 parameter):
• Chain Rule (2 parameters):
• Implicit Differentiation ():
• Implicit Differentiation (): ,
• Linearization:
• Laplace Equation (Harmonic):

Directional Derivatives and Gradients

• Gradient:
• Directional Derivative: (where )
• Maximum rate of increase: in direction

Extreme Values

• Hessian / Discriminant:
• Second Derivative Test: local max if ; local min if ; saddle if

Lagrange Multipliers

• One constraint: and
• Lagrangian:
• Multiple constraints: and

Gradient Descent

• Update rule:

Taylor’s Formula (Two Variables)

• Quadratic approximation at : , where ,

Coordinate Systems

Polar and Cylindrical Coordinates

• Polar ↔︎ Cartesian (2D): , ; ,
• Cylindrical ↔︎ Cartesian (3D): , , ;

Spherical Coordinates

• Spherical ↔︎ Cartesian: , , ;
• Spherical ↔︎ Cylindrical: ,

3D Geometry

• Distance formula:
• Sphere:
• Line (parametric): , ,
• Plane: , normal

Double Integrals

Definition and Fubini’s Theorem

• Double integral (definition):
• Fubini (rectangle):
• Fubini (vertically simple):
• Fubini (horizontally simple):
• Area:
• Average value:

Change of Variables

• Jacobian (2D):
• Change of variables:
• Polar coordinates (Jacobian ):
• Area in polar:
• Gaussian integral:

Triple Integrals

Definition and Fubini’s Theorem

• Triple integral:
• Fubini (general):
• Volume:
• Average value:

Change of Variables

• Jacobian (3D): (3×3 determinant)
• Change of variables:
• Cylindrical coordinates (Jacobian ):
• Spherical coordinates (Jacobian ):

Lagrange Multipliers

• Lagrange condition (2D): , i.e., , ,
• Lagrange function:
• Lagrange condition (3D): , , ,

Taylor’s Formula for Multivariable Functions

• Second degree Taylor polynomial at : , derivatives at
• Maclaurin polynomial (degree 2): , derivatives at
• General degree term: , where ,