Common Anti-Derivatives

There is an art to computing the anti-derivatives of complicated expressions. However, as with most artistic endeavors, there are basic building blocks used to create these works of art, and some basic principles. Recall from your earlier encounters with calculus that:

• If c does not vary with x then

  $$\frac{d}{dx} c = 0.$$

  For example,

  $$\frac{d}{dx} \pi = 0.$$

• For all real numbers x

  $$\frac{d}{dx} x = 1.$$

• If r is a rational number (ratio of integers) and $r \neq 0\; {\rm or}\; 1$then

  $$\frac{d}{dx} x^r = rx^{r-1}$$

  so long as the right hand side is defined for the given value of x. For example,

  $$\frac{d}{dx} x^9 = 9x^8$$

  for all $x\in(-\infty,\infty)$,

  $$\frac{d}{dx} x^{3/2} = \frac{3}{2}x^{1/2}$$

  for $x\geq 0$, and

  $$\frac{d}{dx} x^{1/3} = \frac{1}{3}x^{-2/3}$$

  for $x\in (-\infty,0)\cup(0,\infty)$.
• The chain rule. If g(a) = b, f is differentiable at b, g is differentiable at a and h(x) = f(g(x)), then h is differentiable at a and

  $$h^\prime(a) = f^\prime(b)g^\prime(a) = f^\prime(g(a))g^\prime(a).$$

  For example, if $f(x) = \sin(x)$ and g(x) = 2x then $h(x) = \sin(2x)$ and

  $$h^\prime(a) = \cos(2a)\cdot 2 = 2\cos(2a).$$

• The linear property. If f and g are differentiable at a, c does not depend on x and h(x) = cf(x) + g(x) then h is differentiable at a and

  $$h^\prime(a) = cf^\prime(a) + g^\prime(a)$$

  This rule is frequently applied where either c = 1 or g(x) = 0 for any x. For example, if

  $$h(x) = \frac{\cos(x)}{7} = \frac{1}{7}\cos(x)$$

  then

  $$h^\prime(x) = \frac{1}{7}(-\sin(x)) =-\frac{1}{7}\sin(x).$$

• The product rule. If f and g are differentiable at a and

  h(x) = f(x)g(x)

  then h is differentiable at a and

  $$h^\prime(a) = f^\prime(a)g(a) + f(a)g^\prime(a).$$

  For example, if f(x) = x² and $g(x) = \tan(x)$, then $h(x) = x^2\tan(x)$ and

  $$h^\prime(a) = 2a\tan(a) + a^2\sec^2(a).$$

• The quotient rule. If f and g are differentiable at a, $g(a)\neq 0$ and

  $$h(x) = \frac{f(x)}{g(x)}$$

  then h is differentiable at a and

  $$h^\prime(a) = \frac{f^\prime(a)g(a)-f(a)g^\prime(a)}{g(a)^2}$$

  For example, if $f(x) = \sin(x)$ and $g(x) = \cos(x)$ then $h(x) = \tan(x)$ and

  $$h'(a) = \frac{\cos(a)\cos(a) - \sin(a)(-\sin(a))}{\cos(a)^2} = \frac{1}{\cos(a)^2} = \sec^2(a)$$

Below we have a table of common anti-derivatives. If you read the table from right to left, it is just a table of derivatives, many of which are familiar to you. Others we will learn as we go along, but we record them all here, in case you may need them in your other work before we get to them.

Please find a below a brief table of antiderivatives. In every case, we have ignored any additive constants.

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