The Path of Knowledge and Analysis
This article explains Rolle’s Theorem, one of the fundamental results in differential calculus. The theorem states that if a function is continuous on a closed interval, differentiable on the open interval, and takes equal values at the endpoints, then there exists at least one point inside the interval where the derivative is zero. This theorem forms the foundation for the Mean Value Theorem and many other important results in mathematical analysis.
This article explains the Mean Value Theorem, one of the most important results in differential calculus. The theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point where the slope of the tangent line equals the slope of the secant line connecting the endpoints. This theorem forms the foundation for many applications in mathematics, physics, and engineering.
This article explains Cauchy’s Mean Value Theorem, the generalized form of the Mean Value Theorem. It states that for two continuous and differentiable functions, there exists a point in the interval where the ratio of their derivatives equals the ratio of their overall changes. This theorem is fundamental in proving important results such as L'Hôpital’s Rule and plays a key role in mathematical analysis.
A definite integral is one of the fundamental concepts in calculus, used to compute the accumulated value of a quantity over an interval. This article explains the definition of definite integrals, the Riemann interpretation, the Fundamental Theorem of Calculus, methods of evaluation, and real-world applications.
This article provides a comprehensive collection of the most important integral formulas in calculus, including basic integrals, power rules, exponential and logarithmic integrals, trigonometric and inverse trigonometric integrals, hyperbolic functions, integration techniques, and definite integral identities.
Substitution is one of the most powerful and widely used techniques in integration. It simplifies complex integrals by transforming them into easier forms. This article explains the main substitution methods, including basic u‑substitution, trigonometric substitution, hyperbolic substitution, substitution in rational functions, exponential integrals, trigonometric integrals, and definite integrals.
Integration by parts is one of the most important techniques in calculus for solving integrals involving products of functions. This article explains the formula, the logic behind it, how to choose u and dv, common patterns, advanced applications, and fully worked examples.
Partial fraction decomposition is one of the most powerful techniques for integrating rational functions. By breaking a complicated rational expression into simpler fractions, integration becomes straightforward. This article explains when and how to use partial fractions, the different decomposition cases, step‑by‑step procedures, and fully worked examples.
