# Thread: integration of binomial expansion

1. ## integration of binomial expansion

how would you integrate the general binomial expansion??

thanks for and help

2. $\int (a\times x+b)^n \, dx=\frac{(a x+b)^{n+1}}{a(n+1)}$

If $F (x) = \int f(x)dx$ then:
F(ax+b)=F(x)/a

3. ## thanks

thanks but i was more asking how you would integrate the right hand side of the equation? sorry

4. Originally Posted by hmmmm
thanks but i was more asking how you would integrate the right hand side of the equation? sorry
Since the sum is finite you may interchange the summation and the integral so $\int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^k=\sum_{k=0}^{n}\int\left\{{n\choose{k}}x^{n-k}y^k\right\}$

5. ## thanks

so how do we integrate this?

6. $\sum_{k=0}^{n} {n\choose{k}}x^{n-k}y^k$

$= x^n+nx^{n-1}y+\frac{n(n-1)}{2!}x^{n-2}y^2+\cdots +\frac{n(n-1)\cdots(n-r+1)}{r!}x{n-r}y^r+\cdots+y^n$

So taking:

$\int\sum_{k=0}^{n}{n\choose{k}}x^{n-k}y^kdx$

Is just a case of using the addition and power rules for integrals on this series.
{"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": 6, "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, "math_score": 0.9844309091567993, "perplexity": 1147.4821492215533}, "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}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218203515.32/warc/CC-MAIN-20170322213003-00440-ip-10-233-31-227.ec2.internal.warc.gz"}