Tables


Table 1: Derivatives

The differentiation formulas are listed below. References to the appropriate frames are given. In the following expressions ln x is the natural logarithm or the logarithm to the base e; u and v are variables that depend on x; w depends on u, which in turn depends on x; and a and n are constants. All angles are measured in radians.

  1. StartFraction italic d a Over italic d x EndFraction equals 0
  2. StartFraction d Over italic d x EndFraction left-parenthesis italic a x right-parenthesis equals a
  3. StartFraction italic d x Superscript n Baseline Over italic d x EndFraction equals italic n x Superscript n minus 1
  4. StartFraction d Over italic d x EndFraction left-parenthesis u plus v right-parenthesis equals StartFraction italic d u Over italic d x EndFraction plus StartFraction italic d v Over italic d x EndFraction
  5. StartFraction d Over italic d x EndFraction left-parenthesis italic u v right-parenthesis equals u StartFraction italic d v Over italic d x EndFraction plus v StartFraction italic d u Over italic d x EndFraction
  6. StartFraction d Over italic d x EndFraction left-parenthesis StartFraction u Over v EndFraction right-parenthesis equals StartFraction 1 Over v squared EndFraction left-parenthesis v StartFraction italic d u Over italic d x EndFraction minus u StartFraction italic d v Over italic d x EndFraction right-parenthesis
  7. StartFraction italic d w Over italic d x EndFraction equals StartFraction italic d w Over italic d u EndFraction StartFraction italic d u Over italic d x EndFraction
  8. StartFraction italic d u Superscript n Baseline Over italic d x EndFraction equals italic n u Superscript n minus 1 Baseline StartFraction italic d u Over italic d x EndFraction
  9. StartFraction d ln x Over italic d x EndFraction equals StartFraction 1 Over x EndFraction
  10. StartFraction italic d e Superscript x Baseline Over italic d x EndFraction equals e Superscript x
  11. StartFraction italic d a Superscript x Baseline Over italic d x EndFraction equals a Superscript x Baseline ln a
  12. StartFraction italic d u Superscript v Baseline Over italic d x EndFraction equals italic v u Superscript v minus 1 Baseline StartFraction italic d u Over italic d x EndFraction plus u Superscript v Baseline ln u StartFraction italic d v Over italic d x EndFraction
  13. StartFraction d sine x Over italic d x EndFraction equals cosine x
  14. StartFraction d cosine x Over italic d x EndFraction equals minus sine x
  15. StartFraction d tangent x Over italic d x EndFraction equals secant squared x
  16. StartFraction d secant x Over italic d x EndFraction equals secant x tangent x
  17. StartFraction d cotangent x Over italic d x EndFraction equals minus cosecant squared x
  18. StartFraction d sine Superscript negative 1 Baseline x Over italic d x EndFraction equals StartFraction 1 Over StartRoot 1 minus x squared EndRoot EndFraction
  19. StartFraction d cosine Superscript negative 1 Baseline x Over italic d x EndFraction equals StartFraction negative 1 Over StartRoot 1 minus x squared EndRoot EndFraction
  20. StartFraction d tangent Superscript negative 1 Baseline x Over italic d x EndFraction equals StartFraction 1 Over 1 plus x squared EndFraction
  21. StartFraction d cotangent Superscript negative 1 Baseline x Over italic d x EndFraction equals StartFraction negative 1 Over 1 plus x squared EndFraction

Table 2: Integrals

In the following u and v are variables that depend on x; w is a variable that depends on u, which in turn depends on x; a and n are constants; and the arbitrary integration constants are omitted for simplicity.

  1. integral a italic d x equals italic a x
  2. integral italic a f left-parenthesis x right-parenthesis italic d x equals a integral f left-parenthesis x right-parenthesis italic d x
  3. integral left-parenthesis u plus v right-parenthesis italic d x equals integral u italic d x plus integral v italic d x
  4. integral x Superscript n Baseline italic d x equals StartFraction x Superscript n plus 1 Baseline Over n plus 1 EndFraction comma n not-equals negative 1
  5. integral StartFraction italic d x Over x EndFraction equals ln StartAbsoluteValue x EndAbsoluteValue
  6. integral StartFraction italic d x Over a plus italic b x EndFraction equals StartFraction 1 Over b EndFraction ln StartAbsoluteValue a plus italic b x EndAbsoluteValue
  7. integral e Superscript x Baseline italic d x equals e Superscript x
  8. integral e Superscript italic a x Baseline italic d x equals StartFraction e Superscript italic a x Baseline Over a EndFraction
  9. integral b Superscript italic a x Baseline italic d x equals StartFraction b Superscript italic a x Baseline Over a ln b EndFraction
  10. integral ln StartAbsoluteValue x EndAbsoluteValue italic d x equals x ln StartAbsoluteValue x EndAbsoluteValue minus x
  11. integral sine x italic d x equals minus cosine x
  12. integral cosine x italic d x equals sine x
  13. integral tangent x italic d x equals minus ln StartAbsoluteValue cosine x EndAbsoluteValue
  14. integral cotangent x italic d x equals ln StartAbsoluteValue sine x EndAbsoluteValue
  15. integral secant x italic d x equals ln StartAbsoluteValue secant x plus tangent x EndAbsoluteValue
  16. integral sine x cosine x italic d x equals one half sine squared x
  17. integral StartFraction italic d x Over a squared plus x squared EndFraction equals StartFraction 1 Over a EndFraction tangent Superscript negative 1 Baseline StartFraction x Over a EndFraction
  18. integral StartFraction italic d x Over StartRoot a squared minus x squared EndRoot EndFraction equals sine Superscript negative 1 Baseline StartFraction x Over a EndFraction
  19. integral StartFraction italic d x Over StartRoot x squared plus-or-minus a squared EndRoot EndFraction equals ln StartAbsoluteValue x minus StartRoot x squared plus-or-minus a squared EndRoot EndAbsoluteValue
  20. integral w left-parenthesis u right-parenthesis italic d x equals integral w left-parenthesis u right-parenthesis StartFraction italic d x Over italic d u EndFraction italic d u
  21. integral u italic d v equals italic u v minus integral v italic d u
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
18.222.69.152