Integrals and Areas
Power Rule of Integration
Gives the indefinite integral of any power of x except negative one.
Integral of Reciprocal Function
Integrates the reciprocal function to the natural logarithm of absolute value.
Integral of Exponential Functions
Gives the antiderivatives of natural and general exponential functions.
Integrals of Sine and Cosine
Gives the fundamental antiderivatives of sine and cosine functions.
Integrals of Sec²x and Cosec²x
Gives the standard antiderivatives resulting in tangent and negative cotangent.
Integrals of Sec x Tan x and Cosec x Cot x
Gives the standard antiderivatives yielding secant and negative cosecant.
Integrals of Tan x and Cot x
Gives the logarithmic antiderivatives of the tangent and cotangent functions.
Integrals of Sec x and Cosec x
Gives the logarithmic antiderivatives of the secant and cosecant functions.
Integration by Substitution
Transforms an integral into a simpler variable via the chain rule in reverse.
Integration by Parts
Integrates a product of two functions using the product rule of differentiation.
Integral of eˣ(f(x) + f'(x))
Evaluates an exponential multiplied by the sum of a function and its derivative.
Integral of 1/(x² + a²)
Standard inverse tangent antiderivative for quadratic rational functions.
Integrals of 1/(x² - a²) and 1/(a² - x²)
Logarithmic antiderivatives for differences of squares via partial fractions.
Integral of 1/√(a² - x²)
Standard inverse sine antiderivative for square root differences.
Integral of 1/√(x² ± a²)
Standard logarithmic form for the reciprocal of the square root of x squared plus or minus a squared.
Integral of √(a² - x²)
Standard algebraic-inverse trigonometric integral used frequently in finding circular and elliptical areas.
Integral of √(x² ± a²)
Standard algebraic-logarithmic antiderivative for quadratic radicals under square roots.
Fundamental Theorem of Calculus
Evaluates a definite integral as the difference of values of an antiderivative at the limits.
Additivity of Definite Integrals
Splits a definite integral across adjacent sub-intervals at any intermediate point c.
King's Property of Definite Integrals
Reflects the integrand about the midpoint of the integration interval without changing the value.
Definite Integral of Even and Odd Functions
Simplifies definite integrals over symmetric intervals [-a, a] using the symmetry of the integrand.
Queen's Property (Halving the Upper Limit)
Halves the integration interval [0, 2a] depending on the symmetry of f(2a - x).
Leibniz Rule for Differentiation under Integral
Differentiates an integral with respect to x when the integration limits are functions of x.
Definite Integral as Limit of a Sum
Converts an infinite Riemann sum into a definite integral from 0 to 1.
Area Under a Curve
Calculates the total geometric area bounded by the curve y = f(x), the x-axis, and vertical lines x = a and x = b.
Area Bounded Between Two Curves
Calculates the enclosed area between two intersecting or bounded curves y = f(x) and y = g(x).