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Common Integrals and Derivatives in Calculus

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  • Derivative of \(x^n\)
    The derivative is \(nx^{n-1}\) where n is any real number.
  • Integral of \(x^n\)
    The integral is \(\frac{x^{n+1}}{n+1} + C\) for n \(\neq\) -1.
  • Derivative of \(\sin x\)
    The derivative is \(\cos x\).
  • Integral of \(\sin x\)
    The integral is \(-\cos x + C\).
  • Derivative of \(\cos x\)
    The derivative is \(-\sin x\).
  • Integral of \(\cos x\)
    The integral is \(\sin x + C\).
  • Derivative of \(e^x\)
    The derivative is \(e^x\).
  • Integral of \(e^x\)
    The integral is \(e^x + C\).
  • Derivative of \(\ln x\)
    The derivative is \(\frac{1}{x}\) for x > 0.
  • Integral of \(\frac{1}{x}\)
    The integral is \(\ln|x| + C\) for x \(\neq\) 0.
  • Derivative of \(\tan x\)
    The derivative is \(\sec^2 x\).
  • Integral of \(\sec^2 x\)
    The integral is \(\tan x + C\).
  • Derivative of \(\sec x\)
    The derivative is \(\sec x \tan x\).
  • Integral of \(\sec x \tan x\)
    The integral is \(\sec x + C\).
  • Derivative of \(\arcsin x\)
    The derivative is \(\frac{1}{\sqrt{1 - x^2}}\) for |x| < 1.
  • Integral of \(\frac{1}{\sqrt{1 - x^2}}\)
    The integral is \(\arcsin x + C\).
  • Derivative of \(\arctan x\)
    The derivative is \(\frac{1}{1 + x^2}\).
  • Integral of \(\frac{1}{1 + x^2}\)
    The integral is \(\arctan x + C\).
  • Derivative of constant function
    The derivative of any constant c is \(0\).
  • Integral of constant function \(c\)
    The integral is \(cx + C\).