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~~Calculus~~

Calculus TOC Calculus Differentiation~~Integration~~

Integral Rules of Integration~~Applications of Integration~~

Interpretation~~Geometrical~~Physical Approximation

Net Bounded Area of a Function Average Value of a Function~~Anti-derivative of a Function~~

`-=[]⟨⟩\;',./~!@#$%^&*()_+{}|:"<>? 𝑎𝑏𝑐𝑑𝑒𝑓𝑔ℎ𝑖𝑗𝑘𝑙𝑚𝑛𝑜𝑝𝑞𝑟𝑠𝑡𝑢𝑣𝑤𝑥𝑦𝑧 Å − × ⋅∓±∘꞊﹦∗∙ ℯ 𝔸𝔹ℂ𝔻𝔼𝔽𝔾ℍ𝕀𝕁𝕂𝕃𝕄ℕ𝕆ℙℚℝ𝕊𝕋𝕌𝕍𝕎𝕏𝕐ℤ𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻𝐼𝐽𝐾𝐿𝑀𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈𝑉𝑊𝑋𝑌𝑍 ∼∽∾≁≂≃≄≅≆≇≈≉≌≐≠≡ ≤≥≦≧≨≩≪≫ ∈∉∊∋∌∍ ⊂⊃⊄⊅⊆⊇ 𝛼𝛽𝛾𝛿𝜀𝜁𝜂𝜃𝜄𝜅𝜆𝜇𝜈𝜉𝜊𝜋𝜌𝜎𝜏𝜐𝜑𝜒𝜓𝜔 ∀∂∃∅⦰∆∇∎∞∝∴∵ ∏∐∑⋀⋁⋂⋃ ∧∨∩∪ ∫∬∭∮∯∰∱∲∳ ∥⋮⋯⋰⋱ ‖ ′ ″ ‴ ⁄ ⁗ ʹ ʺ ‵ ‶ ‷ ﹁﹂﹃﹄︹︺︻︼ ︗ ︘ ︿﹀︽︾﹇﹈︷︸ ⏜ ⏝ ⎴ ⎵ ⏞ ⏟ ⏠ ⏡ ←↑→↓↤↦↥↧↔↕↖↗↘↙▲▼◀▶↺↻⟲⟳ ↼↽↾↿⇀⇁⇂⇃⇄⇅⇆⇇ ⇐⇑⇒⇓⇔⇌⇍⇏⇕⇖⇗⇘⇙⇙⇳⥢⥣⥤⥥⥦⥧⥨⥩⥪⥫⥬⥭⥮⥯

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Content


    Anti-Derivative of a Function
 
    Anti-Derivative of simple function  

  
        Indefinite integrals of Derivatives of Polynomials
  
        Indefinite integrals of Derivatives of Trigonometric Functions 
  
        Indefinite integrals of Derivatives of Exponential and Logarithmic Functions 
  
        Indefinite integrals of Derivatives of Inverse Trigonometric Functions

Anti-Derivative of a Function

When considering the average value of a function on a small closed interval Δx, there also exist a mean function value f(ξ) lying on the curve of the function f(x). When the length of subintervals approaching zero, the mean function value of a function on the closed interval Δx will be approaching to f(x) also. Therefore the mean function value of the definite integral of a function f(x) on an infinitesimal interval, Δx is equal to the value of function f(x). By expanding the definite integral of the function f(x) on an infinitesimal interval, Δx, the mean function value f(ξ) is then equal to the change of the integral with respect to x. When taking the limit as Δx tends to zero, function f(x) can therefore be interpreted as the differentiation of the indefinite integral. Imply