
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 16th, 2016, 11:27 PM  #1 
Newbie Joined: Jan 2016 From: sweden Posts: 11 Thanks: 0  Integral: Fundamental theorem of calculus
Hi, I have to solve this integral with the Fundamental theorem of calculus: $\displaystyle f(x)=\int_{\frac{2}{\sqrt{x}}}^{\frac{x^2}{16}} \frac{e^{u}}{u}du$, where $\displaystyle x_{0}=4$ With the help of a similar example, I have done as following, but I am not sure whether I have done right: $\displaystyle \int_{\frac{2}{\sqrt{x}}}^{C}\left(\frac{e^{u}}{u}\right)du + \int_{C}^{\frac{x^2}{16}}\left(\frac{e^{u}}{u}\right)du = \int_{C}^{\frac{2}{\sqrt{x}}}\left(\frac{e^{u}}{u}\right)du+\int_{C}^{\frac{x^2}{16}} \left(\frac{e^{u}}{u}\right)du $ $\displaystyle F'(x)=\frac{e^{\frac{2}{\sqrt{x}}}}{\frac{2}{\sqrt{x}}}\left ( \frac{1}{x^{\frac{3}{2}}} \right )+\frac{e^{\frac{x^{2}}{16}}}{\frac{x^{2}}{16}}\left ( \frac{x}{8} \right )=\frac{e^{\frac{2}{\sqrt{x}}}+4e^{\frac{x^{2}}{16}}}{2x} $ Since: $\displaystyle x_{0}=4$ Answer: $\displaystyle \frac{e^{1}+4e^{1}}{8} $ Well, I have skipped some steps here when I calculated $\displaystyle F'(x)$ because of too many codings. But is my answer right? If not, can I get some hints? Thank you so much. Last edited by skipjack; January 17th, 2016 at 05:01 AM. 
January 17th, 2016, 12:30 AM  #2 
Math Team Joined: Nov 2014 From: Australia Posts: 686 Thanks: 243 
Your question seems to be unrelated to the solution. Is the following question correct? If $\displaystyle f(x) = \int^{x^2/16}_{2/\sqrt{x}}\dfrac{e^{u}}{u}\,du$, find $f'(4)$. If that's the correct question, then your working is spot on. 
January 17th, 2016, 12:35 AM  #3 
Newbie Joined: Jan 2016 From: sweden Posts: 11 Thanks: 0 
Oh sorry, this is the question: Find the Taylor polynomial of order 1 for the function $\displaystyle f$ about the point $\displaystyle x_{0}$. 
January 17th, 2016, 04:18 AM  #4 
Math Team Joined: Jul 2011 From: Texas Posts: 2,761 Thanks: 1416 
$P(x)=f(4)+f'(4) \cdot (x4) = 0 + \dfrac{5}{8e}(x4)$

January 17th, 2016, 05:03 AM  #5 
Newbie Joined: Jan 2016 From: sweden Posts: 11 Thanks: 0 
Yes! Now i understand it. Thank you.


Tags 
calculus, fundamental, integral, theorem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Fundamental theorem of calculus for double integral  Jhenrique  Calculus  6  April 12th, 2014 08:34 PM 
Fundamental theorem of calculus  layd33foxx  Calculus  3  December 12th, 2011 07:32 PM 
Fundamental Theorem of Calculus  Aurica  Calculus  1  June 14th, 2009 08:04 AM 
Fundamental Theorem of Calculus  Aurica  Calculus  1  June 10th, 2009 05:39 PM 
Fundamental Theorem of Calculus  mrguitar  Calculus  3  December 9th, 2007 01:22 PM 