Hyperbolic Functions

Definitions and Properties

Hyperbolic functions are analogues of trigonometric functions, defined using exponential functions. The six standard hyperbolic functions are:

• sinh x:

• cosh x:

• tanh x:

• sech x:

• coth x:

• cosech x:

Key identities include:

Derivatives:

Inverse hyperbolic functions can be expressed in terms of logarithms:

Integrals:

Integration Theory and Applications

Riemann Sums and Area Approximation

Riemann sums approximate the area under a curve by summing areas of rectangles:

• Partition into subintervals of width

• Sample points in each subinterval

• Riemann sum:

Lower and upper sums use minimum and maximum values in each subinterval:

• Lower sum:

• Upper sum:

The Definite Integral

The definite integral is defined as the limit of Riemann sums:

If is continuous and non-negative, the definite integral represents the area under .

Fundamental Theorem of Calculus (FTC2)

• If is an antiderivative of on , then

Properties of the Definite Integral

• If on , then

• If on , then

Integration of Even and Odd Functions

• If is even:

• If is odd:

Mean Value Theorem for Integrals

• There exists such that

• Average value:

Area Between Two Curves

• For on :

• For curves and :

• If curves cross, split the interval and sum the absolute differences

Solids of Revolution

Disk Method

Used to compute the volume of a solid formed by revolving a region about an axis:

• About x-axis:

• About y-axis:

Washer Method

Used when the region has a hole:

Method of Slicing

For solids with known cross-sectional area :

Definite Integral as a Function

Fundamental Theorem of Calculus 1 (FTC1)

• If is continuous on , then is an antiderivative of

Advanced Integration Techniques

Inverse Trigonometric, Exponential, and Logarithmic Integrals

Integration by Parts (IBP)

• Reduction formulas for powers and logarithms

Trigonometric Substitutions

Trigonometric Integrals

• Use identities for powers of sine and cosine

• Products of powers: use substitution or reduction formulas

• Integrals involving and :

Partial Fraction Decomposition

Used to integrate rational functions:

• If degree of numerator denominator: perform long division

• Decompose denominator into linear and irreducible quadratic factors

• Set up partial fractions and solve for constants

Improper Integrals

Type I: Infinite Interval

• Converges if limit exists and is finite

Type II: Infinite Discontinuity

• improper if has infinite discontinuity at or

• Converges if limit exists

p-Test for Improper Integrals

• converges if , diverges if

Sequences and Series

Sequences

• A sequence is an ordered list

• Convergent if

• Monotonic: increasing or decreasing

• Bounded: exists such that

• Monotonic Sequence Theorem: every bounded, monotonic sequence converges

Series

• A series is the sum

• Convergent if sequence of partial sums converges

• Geometric series: converges if

• Test for divergence: if , series diverges

Convergence Tests

• Integral Test: compare series to improper integral

• p-Series Test: converges if

• Comparison Test: compare to known convergent/divergent series

• Alternating Series Test: converges if terms decrease to zero

• Absolute Convergence: if converges, so does

• Ratio Test: ; converges if , diverges if

• Root Test: ; converges if , diverges if

Power Series

Definition and Convergence

• Power series:

• Radius of convergence ; converges for

• Interval of convergence: may include or exclude endpoints

Term-by-Term Differentiation and Integration

• If converges for :

Common Maclaurin Series

Binomial Series

General Form

• For , :

• Binomial coefficient:

Differential Equations

Types and Solutions

• Order: highest derivative present

• Separable ODE: can be written as

• Linear ODE: ; solved using integrating factor

• Homogeneous ODE: ; substitution or reduces to separable

• Exact ODE: is exact if

Example: Solve (separable):

Example: Linear ODE :

• Integrating factor

• General solution:

Additional info: This guide covers all major topics from the second semester calculus curriculum, including advanced integration, sequences and series, power series, and introductory differential equations. Worked examples and tutorial questions are provided throughout for practice.