# Problem with understanding integration by parts method?

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 $a$

$a$ 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.
• 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 $\ln(-1)$? – Fly by Night Jul 6 '13 at 22:56
• @FlybyNight, sure, but if $a\le 0$, then $\ln a$ is undefined, so the integral formula doesn't apply anyways. – vadim123 Jul 6 '13 at 22:57