# Plz explain how to integrate this function

• Jul 31st 2011, 06:00 PM
nicksbyman
Plz explain how to integrate this function
I don't understand how they go from the second to last step to the last step. Those two equations aren't equal unless the Cs are different. If they are please explain this.

Thanks a bunch :D

P.S. sorry if the image is a little blurry. If it is please say so in the thread and I will re-screenshot the window with the entire window selected so it is not as blurry.
• Jul 31st 2011, 06:16 PM
Siron
Re: Plz explain how to integrate this function
They're doing a substitution (better: rewriting the integral). But you can also do this: you can calculate $\int 0,4^{x}dx$ directly, but because the exponent is now $\frac{x}{3}$ you've to do a substitution, let $\frac{x}{3}=t$ then $dx=3\cdot dt$ and so the integral becomes:
$3 \int 0,4^{t}dt$, you can calculate this integral directly, they do the same, because you can write:
$3 \int 0,4^{\frac{x}{3}}\left(\frac{1}{3}dx\right)=\int 0,4^{\frac{x}{3}}\left(\3\cdot \frac{1}{3}dx\right)=\int 0,4^{\frac{1}{3}}dx$
• Jul 31st 2011, 06:21 PM
nicksbyman
Re: Plz explain how to integrate this function
I am pretty sure I understand everything you just said but I don't understand how they get 3/ln.4 *.4^(x/3) + C = 3ln(2.5)*.4^(x/3) + C
• Jul 31st 2011, 06:41 PM
Siron
Re: Plz explain how to integrate this function
Do you know that (in general):
$\int a^{x}dx=\frac{a^x}{\ln(a)}+C$
• Jul 31st 2011, 06:44 PM
nicksbyman
Re: Plz explain how to integrate this function
Yes.
• Jul 31st 2011, 06:45 PM
Siron
Re: Plz explain how to integrate this function
So is it clear now? ...
• Jul 31st 2011, 06:46 PM
nicksbyman
Re: Plz explain how to integrate this function
No, I still have the same confusion I had in post # 3. To possibly make myself clearer: I don't understand why the second to last and last steps in the worked out solution attached are equal.
• Aug 1st 2011, 06:25 AM
Ackbeet
Re: Plz explain how to integrate this function
Quote:

Originally Posted by nicksbyman
No, I still have the same confusion I had in post # 3. To possibly make myself clearer: I don't understand why the second to last and last steps in the worked out solution attached are equal.

They are not equal, and I would say the last step is invalid, though not the only invalid step.

$\frac{1}{\ln(0.4)}\approx -1.09,$ whereas

$\ln(2.5)\approx 0.9163.$

The first is less than zero, and the second greater than zero. They are not equal.

Conclusion: the answer being propounded is incorrect.
• Aug 1st 2011, 10:25 AM
nicksbyman
Re: Plz explain how to integrate this function
Thanks Ackbeet. Just to confirm that there is an error in this solution and I am not simply taking this out of context, I would really appreciate it if someone were to go to this link: CalcChat | Calculus 8e | Easy Access Study Guide and confirm that their solution is erroneous.

Thanks again.
• Aug 1st 2011, 12:18 PM
Ackbeet
Re: Plz explain how to integrate this function
I would say that step 4 is correct, but step 5 is incorrect.
• Aug 2nd 2011, 07:31 AM
Siron
Re: Plz explain how to integrate this function
Thanks Ackbeet to confirm my thoughts, I was thinking and thinking but I couldn't see why step 4 was equal to step 5 so I thought I did something wrong but now you confirm it's wrong so I'm sure :).