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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.

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 $\displaystyle \int 0,4^{x}dx$ directly, but because the exponent is now $\displaystyle \frac{x}{3}$ you've to do a substitution, let $\displaystyle \frac{x}{3}=t$ then $\displaystyle dx=3\cdot dt$ and so the integral becomes:

$\displaystyle 3 \int 0,4^{t}dt$, you can calculate this integral directly, they do the same, because you can write:

$\displaystyle 3 \int 0,4^{\frac{x}{3}}\left(\frac{1}{3}dx\right)=\int 0,4^{\frac{x}{3}}\left(\3\cdot \frac{1}{3}dx\right)=\int 0,4^{\frac{1}{3}}dx$

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

Re: Plz explain how to integrate this function

Do you know that (in general):

$\displaystyle \int a^{x}dx=\frac{a^x}{\ln(a)}+C$

Re: Plz explain how to integrate this function

Re: Plz explain how to integrate this function

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.

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.

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

$\displaystyle \ln(2.5)\approx 0.9163.$

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

Here is the correct answer.

Conclusion: the answer being propounded is incorrect.

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.

Re: Plz explain how to integrate this function

I would say that step 4 is correct, but step 5 is incorrect.

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 :).