# How do you use the Fundamental Theorem of Calculus to find the derivative of int (u^3) / (1+u^2) du from 2-3x to 5?

Feb 17, 2017

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

#### Explanation:

The following assumes you mean the derivative wrt $x$.

$\frac{d}{\mathrm{du}} \left({\int}_{a}^{u} f \left(t\right) \setminus \mathrm{dt}\right) = f \left(u\right)$ where $a =$ some constant.

If in fact $u = u \left(x\right)$, we can use the Chain Rule so that:

$\frac{d}{\mathrm{dx}} \left({\int}_{a}^{u \left(x\right)} f \left(t\right) \setminus \mathrm{dt}\right) = f \left(u \left(x\right)\right) \frac{\mathrm{du}}{\mathrm{dx}}$

We can also play with the intervals ..... in summary ${\int}_{a}^{b} = {\int}_{a}^{0} + {\int}_{0}^{b} = {\int}_{0}^{b} - {\int}_{0}^{a}$.... in order to reach this point:

$\frac{d}{\mathrm{dx}} \left({\int}_{{u}_{1} \left(x\right)}^{{u}_{2} \left(x\right)} f \left(t\right) \setminus \mathrm{dt}\right) = f \left({u}_{2} \left(x\right)\right) \frac{{\mathrm{du}}_{2}}{\mathrm{dx}} - f \left({u}_{1} \left(x\right)\right) \frac{{\mathrm{du}}_{1}}{\mathrm{dx}}$

Gasp! With all that, we just need to process this:

$\frac{d}{\mathrm{dx}} \left({\int}_{2 - 3 x}^{5} \frac{{u}^{3}}{1 + {u}^{2}} \mathrm{du}\right)$

$= \frac{{5}^{3}}{1 + {5}^{2}} \frac{d \left(5\right)}{\mathrm{dx}} - \frac{{\left(2 - 3 x\right)}^{3}}{1 + {\left(2 - 3 x\right)}^{2}} \frac{d \left(2 - 3 x\right)}{\mathrm{dx}}$

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

Maybe you can further simplify that.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8623057,"math_prob":1.0000095,"size":445,"snap":"2021-43-2021-49","text_gpt3_token_len":115,"char_repetition_ratio":0.106575966,"word_repetition_ratio":0.0,"special_character_ratio":0.2494382,"punctuation_ratio":0.13541667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999963,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-22T16:33:24Z\",\"WARC-Record-ID\":\"<urn:uuid:7fc54f46-95b2-476d-81d2-6c45e0d4d0c3>\",\"Content-Length\":\"35169\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d48992de-8933-4180-bf0c-338be518c131>\",\"WARC-Concurrent-To\":\"<urn:uuid:fd76b805-2d99-49f7-9e9f-5712d414e574>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-use-the-fundamental-theorem-of-calculus-to-find-the-derivative-of-int-4#379371\",\"WARC-Payload-Digest\":\"sha1:7MCLHI2ZYY2X6QM24LV6MHORN7LOW2EH\",\"WARC-Block-Digest\":\"sha1:CSFIK6CF4H2JEWNIRSC6LXXOZWDC4Q2Z\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585516.51_warc_CC-MAIN-20211022145907-20211022175907-00331.warc.gz\"}"}