1. ## Integration problem

I am having a problem with this calculus problem:

$\displaystyle \int \frac{1}{y^2-16} dy$

I factored it to: $\displaystyle \int \frac{1}{(y-4)(y+4)} dy$

Then I tried integration by parts: $\displaystyle u=(y-4)^{-1} dv=(y+4)^{-1}$

$\displaystyle du=-(y-4)^{-2} v=ln(y+4)$

$\displaystyle \frac{ln(y+4)}{(y-4)}-\int \frac{ln(y+4)}{(y-4)^2} dy$

So I'm stuck, and I'm just wondering what is the best way to do this?
Peter

2. Originally Posted by flybynight
I am having a problem with this calculus problem:

$\displaystyle \int \frac{1}{y^2-16} dy$

I factored it to: $\displaystyle \int \frac{1}{(y-4)(y+4)} dy$

Then I tried integration by parts: $\displaystyle u=(y-4)^{-1} dv=(y+4)^{-1}$

$\displaystyle du=-(y-4)^{-2} v=ln(y+4)$

$\displaystyle \frac{ln(y+4)}{(y-4)}-\int \frac{ln(y+4)}{(y-4)^2} dy$

So I'm stuck, and I'm just wondering what is the best way to do this?
Peter
use the method of partial fractions ...

$\displaystyle \displaystyle \frac{1}{y^2-16} = \frac{A}{y+4} + \frac{B}{y-4}$

solve for $\displaystyle A$ and $\displaystyle B$ , then integrate each term individually

3. Two more methods:

2. Put $\displaystyle y = \tanh{t}$

3. Let $\displaystyle u = \frac{y-4}{y+4}$.
{"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": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://mathhelpforum.com/calculus/156858-integration-problem.html", "fetch_time": 1527152909000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-22/segments/1526794866107.79/warc/CC-MAIN-20180524073324-20180524093324-00423.warc.gz", "warc_record_offset": 194998223, "warc_record_length": 9678, "token_count": 444, "char_count": 1227, "score": 4.34375, "int_score": 4, "crawl": "CC-MAIN-2018-22", "snapshot_type": "latest", "language": "en", "language_score": 0.6811811327934265, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00037-of-00064.parquet"}