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First of all (before I get started with integration by parts method to solve integrals) I need to know what is the meaning of all terms in integration rules like $a^x,\,e^x,\,\ln\left(x\right),\,$ etc.

I'm having problem with solving integrals using integration by parts method because of integration rules terms meaning, so can any one help me understanding the terms and give me

Examples.

I think list below have most of integration rules terms:

  • $a$ $\to$ I really need to know.
  • $x$ $\to$ I think $x$
  • $e^x$ $\to$ I really need to know.
  • $a^x$ $\to$ I really need to know.
  • $\ln\left(x\right)$ $\to$ I really need to know.

I mean: for the rule $\int a\,\,dx=ax+C$ what is $a$

Please chick this link http://www.mathsisfun.com/calculus/integration-rules.html

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$a$ is any constant, that does not change when $x$ changes. For example, $3$ or $-7$. When they write $\int a dx=ax+C$, that means that $\int 3dx=3x+C$ and $\int (-7)dx=-7x + C$.

$e^x$ is the exponential function, with base $e$.

$a^x$ is the exponential function, with base $a$, where again $a$ is any constant.

$\ln x$ is the (natural) logarithm function.

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  • $\begingroup$ Don't we really need $a > 0$ ? Suppose we let $a = -1$, what would the graph $y = a^x$ look like? The standard trick for integrating $a^x$ is to rewrite as $\operatorname{e}^{x\ln a}$. What is $\ln(-1)$? $\endgroup$ – Fly by Night Jul 6 '13 at 22:56
  • $\begingroup$ @FlybyNight, sure, but if $a\le 0$, then $\ln a$ is undefined, so the integral formula doesn't apply anyways. $\endgroup$ – vadim123 Jul 6 '13 at 22:57
  • $\begingroup$ So that needs to be made clear to the OP. S/he is just starting to learn and ought to be made aware of such things. $\endgroup$ – Fly by Night Jul 6 '13 at 22:59
  • $\begingroup$ Thank you vadim123, now I think I have a basic Informations. $\endgroup$ – Mohammad Fakhrey Jul 6 '13 at 23:00

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