U Substitution

[NaN-Gram]

I got about halfway through my problem, but I messed up. How do I duplicate the question in my answer, in order to divide the equation by two?

 

[Dr.Peterson]

For those who don't want to open a pdf, here is the image as it should have been attached:



I'm not sure I understand your question. What does "duplicate the question" mean?

Can you explain what you did here, and why? The last few lines are hard to follow.

 

[Cubist]

I also don't understand "duplicate the question". But I've looked at your work and spotted a couple of problems.

For question 5, the integral is correct "e^x(sin(x)+cos(x))/2" however when applying the limits you missed the following red minus sign:-

[math] \frac{e^\pi(0-1)}{2} = \color{red}-\color{black} \frac{e^\pi}{2}[/math]
--

For question 6 you make another minus sign mistake, the highlighted "+" below is correct (you wrote "-"):-

[math] \sec^{2}\left(x\right)= \tan^{2}\left(x\right)\color{red}+\color{black}1[/math]
I used to make lots of minus sign mistakes. With practice and some extra caution this will happen less often.

 

[Dr.Peterson]

Ignoring the question of what you did in #6, and how much was wrong in it; how about starting over?

Let u = sec(x) right at the beginning ... do you see du hidden in there?

 

[Steven G]

\(\displaystyle Define \ I \ = \int e^xcos(x) dx\)

\(\displaystyle Let \ u \ = e^x, \ \ du=e^xdx \ NOT \ e^x, \ dv = cos(x)dx, v= sin(x)\)

\(\displaystyle I = e^xsin(x) - \int e^xsin(x)dx\)

\(\displaystyle Let \ u\ = e^x. \ \ du = e^xdx, \ dv = sin(x), v = -cos(x)\)

\(\displaystyle I = e^xsin(x) - [-e^xcos(x) - \int (-e^x)cosxdx] = e^xsin(x) + e^xcos(x) - \int (e^x)cosxdx = e^xsin(x) + e^xcos(x) - I\)

There is the duplication you want. Can you continue?

 

[NaN-Gram]

Dear Dr Peterson, Mr Cubist, and Mr Jomo,

Thank you all for your help with my question. I did mean that I wanted to put the expression from question 5 in the answer, I'm sorry that I was unclear about it earlier.

I still need help with question 6, but I can complete question 5 on paper. Is it okay if I post my solution in this thread?

 

[Steven G]

#6

What is the derivative of sec(x)? That is if u = sec(x), then what will du equal?

 

[Steven G]

I still need help with question 6, but I can complete question 5 on paper. Is it okay if I post my solution in this thread?

Sure

 

[NaN-Gram]

Here's my on paper solution, I hope that this meshes with what you shared.

It applies to question 5.

By the way, the derivative of secx is secxtanx, from what I found.

 

[Steven G]

That answer is not correct. I did all the work for you did you not understand some of it?

You say that you found the derivative of sec(x). What does that mean? Did you spend your valuable time looking it up somewhere or did you take the quick way out and simply derive it?

\(\displaystyle \dfrac{d}{dx}sec(x) = \dfrac{d}{dx}(cos(x))^{-1} = -1(cos(x))^{-2}(-sin(x)) = \dfrac{sin(x)}{cos^2(x)} = \dfrac{sin(x)}{cos(x)}\dfrac{1}{cos(x)} = tan(x)sec(x)\)

Did my hint asking what the derivative of sec(x) help you do your problem? What answer did you get or can you please tell us where you got stuck?

 

[NaN-Gram]

I apologize for my confusion. I wanted to try the problem again, but I didn't read your explanation carefully enough. That was my bad.
I corrected the problem below, you had the right idea to turn the expression into a variable.

Anyway, I'll make a new thread for question 6, seeing as it's a different problem. Thank you everyone!

 

[Steven G]

I apologize for my confusion. I wanted to try the problem again, but I didn't read your explanation carefully enough. That was my bad.
I corrected the problem below, you had the right idea to turn the expression into a variable.
View attachment 17249
Anyway, I'll make a new thread for question 6, seeing as it's a different problem. Thank you everyone!

You have the correct answer. Good job.