# The Unapologetic Mathematician

## Indefinite Integration

Since we’ve established the connection between integration and antidifferentiation, we’ll be concerned mostly with antiderivatives more directly than derivatives. So, it’s useful to have some simple notation for antiderivatives.

That’s pretty much what the “indefinite integral” amounts to. It looks like an integral, and it does (what the FToC tells us is) all the hard work of integration, but it stops short of actually calculating an integral. Given a function $f(x)$, we write an antiderivative as $\int f(x)dx$. Note that we aren’t saying which antiderivative we mean, and for the purposes of the FToC (part 2), we don’t need to be. It’s customary, though, to write the result generically by adding a $+C$ to the end of it.

We know, for example, that

$\displaystyle\frac{d}{dx}\frac{x^{n+1}}{n+1}=\frac{(n+1)x^n}{n+1}=x^n$

Then we turn this around to write

$\displaystyle\int x^ndx=\frac{x^{n+1}}{n+1}+C$

and so on.

We can also go back and rewrite the two rules of integration we found before:

$\displaystyle\int f(x)+g(x)dx=\int f(x)dx+\int g(x)dx$
$\displaystyle\int cf(x)dx=c\int f(x)dx$

Notice here that we don’t need to add the $+C$, since each side consists of indefinite integrals. We can hide these “constants of integration” on both sides. They only need to show up once we fully evaluate an indefinite integral.

### Like this:

Posted by John Armstrong | Analysis, Calculus

## 1 Comment	»

1. [...] is defined for all measurable functions . This isn’t quite the same indefinite integral that we’ve worked with before. In that case we only considered functions , picked a base-point , and defined a new function on [...]

Pingback by | May 27, 2010 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 8, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.892503559589386, "perplexity_flag": "middle"}