Table of Standard Forms

Excerpts

• Table of Standard Forms

  [-12pt]0pt32pt$nx^{n-1}$ & $x^n$ &$ \dfrac{1}{n+1} x^{n+1} + C $

• Table of Standard Forms

  $\epsilon^x$ & $\epsilon^x$ & $\epsilon^x + C$

• Table of Standard Forms

  [-12pt]0pt32pt$u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}$ & $uv$ & No general form known

• Table of Standard Forms

  $x^{-1}$ & $\log_\epsilon x$ & $ x(\log_\epsilon x - 1) + C$

• Table of Standard Forms

  $\sec^2 x$& $\tan x$ & $-\log_\epsilon \cos x + C $

• Table of Standard Forms

  $ 2a·\sin 2ax$ & $\sin^2 ax$ & $\dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C $

Equations

• Table of Standard Forms

  \frac{1}{2} x^2 + C

  The integral of x with respect to x is one half x squared plus the constant of integration (row with dy/dx = 1, y = x).

• Table of Standard Forms

  ax + C

  The integral of the constant a with respect to x is a x plus the constant of integration (row with y = a).

• Table of Standard Forms

  \frac{1}{2} ax^2 + C

  The integral of a x with respect to x is one half a x squared plus the constant of integration.

• Table of Standard Forms

  \frac{1}{3} x^3 + C

  The integral of x squared with respect to x is one third x cubed plus the constant of integration.

• Table of Standard Forms

  \dfrac{1}{n+1} x^{n+1} + C

  The integral of x to the power n is x to the power n+1 divided by n+1, plus the constant of integration.

• Table of Standard Forms

  \log_\epsilon x + C

  The integral of 1/x with respect to x is the natural logarithm of x plus the constant of integration.

• Table of Standard Forms

  \int u\, dx ± \int v\, dx ± \int w\, dx

  The integral of a sum or difference of functions is the sum or difference of their integrals.

• Table of Standard Forms

  u\, \dfrac{dv}{dx} + v\, \dfrac{du}{dx}

  The derivative of the product uv is u times dv/dx plus v times du/dx (the table gives no general integral of this row).

• Table of Standard Forms

  \dfrac{v\, \dfrac{du}{dx} - u\, \dfrac{dv}{dx}}{v^2}

  The derivative of the quotient u/v is (v du/dx minus u dv/dx) divided by v squared.

• Table of Standard Forms

  ux - \int x\, du + C

  The integral of u with respect to x equals u x minus the integral of x du, plus the constant of integration.

• Table of Standard Forms

  \epsilon^x + C

  The integral of the exponential function epsilon to the x is itself plus the constant of integration.

• Table of Standard Forms

  x(\log_\epsilon x - 1) + C

  The integral of the natural logarithm of x with respect to x is x times (log x minus 1), plus the constant of integration.

• Table of Standard Forms

  0.4343x (\log_\epsilon x - 1) + C

  The integral of the common logarithm of x with respect to x is 0.4343 x times (natural log of x minus 1), plus the constant of integration; 0.4343 is the modulus 1/log to base epsilon of 10.

• Table of Standard Forms

  \dfrac{a^x}{\log_\epsilon a} + C

  The integral of a to the power x with respect to x is a^x divided by the natural logarithm of a, plus the constant of integration.

• Table of Standard Forms

  -\cos x + C

  The integral of sin x with respect to x is minus cos x, plus the constant of integration.

• Table of Standard Forms

  \sin x + C

  The integral of cos x with respect to x is sin x, plus the constant of integration.

• Table of Standard Forms

  -\log_\epsilon \cos x + C

  The integral of tan x with respect to x is minus the natural logarithm of cos x, plus the constant of integration.

• Table of Standard Forms

  x · \arcsin x + \sqrt{1 - x^2} + C

  The integral of arcsin x with respect to x is x arcsin x plus the square root of one minus x squared, plus the constant of integration.

• Table of Standard Forms

  x · \arccos x - \sqrt{1 - x^2} + C

  The integral of arccos x with respect to x is x arccos x minus the square root of one minus x squared, plus the constant of integration.

• Table of Standard Forms

  x · \arctan x - \frac{1}{2} \log_\epsilon (1 + x^2) + C

  The integral of arctan x with respect to x is x arctan x minus one half the natural logarithm of one plus x squared, plus the constant of integration.

• Table of Standard Forms

  \cosh x + C

  The integral of sinh x with respect to x is cosh x, plus the constant of integration.

• Table of Standard Forms

  \sinh x + C

  The integral of cosh x with respect to x is sinh x, plus the constant of integration.

• Table of Standard Forms

  \log_\epsilon \cosh x + C

  The integral of tanh x with respect to x is the natural logarithm of cosh x, plus the constant of integration.

• Table of Standard Forms

  \log_\epsilon (x+a) + C

  The integral of 1/(x+a) with respect to x is the natural logarithm of x+a, plus the constant of integration.

• Table of Standard Forms

  \log_\epsilon (x + \sqrt{a^2 + x^2}) + C

  The integral of 1 over the square root of a squared plus x squared is the natural logarithm of x plus that square root, plus the constant of integration.

• Table of Standard Forms

  \dfrac{x}{\sqrt{a^2 + x^2}} + C

  The integral of a squared over (a squared plus x squared) to the three halves is x over the square root of a squared plus x squared, plus the constant of integration.

• Table of Standard Forms

  -\dfrac{1}{a} \cos ax + C

  The integral of sin ax with respect to x is minus cos ax divided by a, plus the constant of integration.

• Table of Standard Forms

  \dfrac{1}{a} \sin ax + C

  The integral of cos ax with respect to x is sin ax divided by a, plus the constant of integration.

• Table of Standard Forms

  -\dfrac{1}{a} \log_\epsilon \cos ax + C

  The integral of tan ax with respect to x is minus one over a times the natural logarithm of cos ax, plus the constant of integration.

• Table of Standard Forms

  \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C

  The integral of sin squared x with respect to x is x over 2 minus sin 2x over 4, plus the constant of integration.

• Table of Standard Forms

  \dfrac{x}{2} + \dfrac{\sin 2x}{4} + C

  The integral of cos squared x with respect to x is x over 2 plus sin 2x over 4, plus the constant of integration.

• Table of Standard Forms

  -\frac{\cos x}{n} \sin^{n-1} x + \frac{n-1}{n} \int \sin^{n-2} x\, dx + C

  Reduction formula: the integral of sin to the power n of x is minus cos x sin to the power n-1 over n, plus (n-1)/n times the integral of sin to the power n-2, plus the constant of integration.

• Table of Standard Forms

  \log_\epsilon \tan \dfrac{x}{2} + C

  The integral of csc x with respect to x is the natural logarithm of tan(x/2), plus the constant of integration.

• Table of Standard Forms

  -\cotan x + C

  The integral of csc squared x with respect to x is minus the cotangent of x, plus the constant of integration.

• Table of Standard Forms

  \log_\epsilon \tan x + C

  The integral of 1 over sin x cos x with respect to x is the natural logarithm of tan x, plus the constant of integration.

• Table of Standard Forms

  \frac{1}{2} \cos(m - n)x - \frac{1}{2} \cos(m + n)x + C

  FLAG (book/extraction discrepancy): the table gives this as the integral of sin mx sin nx. Differentiating the stated result does not return sin mx sin nx (for m = n it gives an x-independent expression), and the correct integral is sin((m-n)x)/(2(m-n)) minus sin((m+n)x)/(2(m+n)). Recorded as printed, not corrected.

• Table of Standard Forms

  \dfrac{x}{2} - \dfrac{\sin 2ax}{4a} + C

  The integral of sin squared ax with respect to x is x over 2 minus sin 2ax over 4a, plus the constant of integration. FLAG: the same row prints the derivative of sin squared ax as 2a sin 2ax; differentiation gives a sin 2ax, so the derivative column disagrees with this integral by a factor of 2. Recorded as printed.

• Table of Standard Forms

  \dfrac{x}{2} + \dfrac{\sin 2ax}{4a} + C

  The integral of cos squared ax with respect to x is x over 2 plus sin 2ax over 4a, plus the constant of integration. FLAG: the same row prints the derivative of cos squared ax as minus 2a sin 2ax; differentiation gives minus a sin 2ax, a discrepancy of a factor of 2 in the derivative column. Recorded as printed.

Problems

No exercises in this chapter.