# How do you integrate int(x+1)/((2x-4)(x+5)(x-2)) using partial fractions?

First we observe that

$\int \frac{x + 1}{\left(2 x - 4\right) \left(x + 5\right) \left(x - 2\right)} \mathrm{dx} = \frac{1}{2} \cdot \int \frac{x + 1}{\left(x + 5\right) \cdot {\left(x - 2\right)}^{2}} \mathrm{dx}$

Hence we have to find constants A,B,C such as

$\frac{x + 1}{\left(x + 5\right) \cdot {\left(x - 2\right)}^{2}} = \frac{A}{x + 5} + \frac{B}{x - 2} + \frac{C}{x - 2} ^ 2$

We can calculate these constants by giving x three different values

for example $x = 0 , x = - 1 , x = 3$

hence we get

$\frac{x + 1}{\left(x + 5\right) \cdot {\left(x - 2\right)}^{2}} = - \frac{2}{49 \cdot \left(x + 5\right)} + \frac{2}{49 \cdot \left(x - 2\right)} + \frac{3}{14 {\left(x - 2\right)}^{2}}$

Hence now we have that

int(x+1)/((2x-4)(x+5)(x-2))dx=1/2*int (x+1)/[(x+5)*(x-2)^2]dx= 1/2*[int (-2)/(49*(x+5))dx +int 2/(49(x-2))dx+int 3/[(14)(x-2)^2]dx]= 1/2*[-2/49*ln(x+5)+2/49*ln(x-2)-3/14*(1/(x-2))]+c
{"ft_lang_label":"__label__en","ft_lang_prob":0.73349,"math_prob":1.00001,"size":407,"snap":"2021-43-2021-49","text_gpt3_token_len":110,"char_repetition_ratio":0.09181141,"word_repetition_ratio":0.0,"special_character_ratio":0.24815725,"punctuation_ratio":0.049382716,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000019,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-24T15:06:44Z\",\"WARC-Record-ID\":\"<urn:uuid:c031a9b6-59c1-41b2-aa48-986dfebd8c73>\",\"Content-Length\":\"34085\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:98186b5f-4696-46a4-9d07-d4e4492ca1aa>\",\"WARC-Concurrent-To\":\"<urn:uuid:11a9a43a-7564-439a-8bf7-78c1809e5309>\",\"WARC-IP-Address\":\"142.251.45.116\",\"WARC-Target-URI\":\"https://api-project-1022638073839.appspot.com/questions/how-do-you-integrate-int-x-1-2x-4-x-5-x-2-using-partial-fractions\",\"WARC-Payload-Digest\":\"sha1:CNMLXHIFHUNNSCSTRCW4HM7ZC7PMMPKU\",\"WARC-Block-Digest\":\"sha1:MAQTNWNYVK75RRYKHRO4A72L6M6P3JVE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323586043.75_warc_CC-MAIN-20211024142824-20211024172824-00635.warc.gz\"}"}