integration qstn

[torpid pine]

can someone pls solve this and tell me how to think? I tried simplifying ln (x/e)^x
but I cant understand which part is supposed to be the first term or 2nd term for integration by parts

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[torpid pine]

it becomes so complicated that it's difficult to keep track of things

I got stuck here

[marble valve]

Note that (a^x f(x))' = a^x (f'(x) + f(x) ln(a)).

What I mean is that splitting the integral into several terms probably won't help.

[swift portal]

Load mat le option differentiate kar

[torpid pine]

but then how do I solve it

[torpid pine]

tried that. option b worked. that's the ans too. the thing is do I need to differentiate the option every time when I face these kind of problems? I mean ye theyre hard. but what if its a numerical one? I want to know the method of solving it and ugh how to approach these... for other questions like this. also checking each option takes a lotta time

[marble valve]

Use the property that I wrote above.

According to it, we should have f(x) = ln((x/e)^x), f'(x) = ln(x). So, verify that. If it does work, then the antiderivative comes out right away.

[swift portal]

11th ko hai kya 12th ko

[marble valve]

What?

[torpid pine]

damn thats genius fx = x/e)^x so ans is a^x x/e)^x

i was trying to use some other way... like actually manipulating stuff inside it

became pretty complicated

[torpid pine]

is there a way to solve it using some manipulation?

[torpid pine]

12th

[torpid pine]

like using by parts

[marble valve]

I think you missed some symbols. If you meant that the antiderivative is a^x ln((x/e)^x), then that's correct.

[torpid pine]

yep sorry I was on my laptop that 9 key doesn't work there

[marble valve]

Oh, ok.

[torpid pine]

I was trying to use by parts..

even saw a solution in the internet that used by parts but it was pretty long and complicated so I uhm couldn't follow it yk

[torpid pine]

what do u think @marble valve

[marble valve]

Oh, can you show it?

[torpid pine]

sure just a min

lost track of things in that by parts step... how am I supposed to know which one has to be the first term in these complex questions

[marble valve]

Ah, I think I get the general idea.
However, instead of rewriting the first step, we can just split into two terms right away:
∫(a^x (ln(x) + ln(a) ln((x/e)^x))dx) = ∫(a^x ln(x)dx) + ln(a)∫(a^x ln((x/e)^x)dx)
And now try integration by parts for the second term: u = ln((x/e)^x, dv = ln(a) a^x.

[torpid pine]

but what about the first term? I mean the lnx . a^x...when I try to use by parts it becomes something like
lnx ∫a^x dx - ∫( d(lnx)/dx ∫a^x dx )dx
on simplifying I get the 2nd term as ∫ a^x /(x lnx) dx

this

how do I solve this

[fading geyser]

I’m not quite sure that the integral of a^x is a^x/ln(x).

In fact it is a^x/ln(a).

Then you can use this edit to notice what appears in the integral.

[outer radish]

this might just be a non-elementary function 💀

but yeah $\int a^xdx=\frac{a^x}{\ln(a)}+C$

[iron rock]

ooaa

[outer radish]

but still you're dealing with non-elementary function

[marble valve]

Don't do anything to it. Work with the second term and you'll see what happens.

[torpid pine]

oh yeah right. my bad. it becomes a^x / x
do I need to use by parts again?

[torpid pine]

kocher is right. it should be a^x/x

[torpid pine]

this?

[marble valve]

The last line is wrong. Not sure how you got that.

Oh, the second to last line also isn't quite right.

We have:
u = ln((x/e)^x, dv = ln(a) a^x dx
du = ln(x)dx, v = a^x
So:
ln(a)∫(a^x ln((x/e)^x)dx) = a^x ln((x/e)^x - ∫(a^x ln(x)dx)
But we also had the first term. So:
∫(a^x ln(x)dx) + ln(a)∫(a^x ln((x/e)^x)dx) = ∫(a^x ln(x)dx) + a^x ln((x/e)^x - ∫(a^x ln(x)dx) = a^x ln((x/e)^x + C
So, that's the result.

[fading geyser]

Yeah, do what you want there, I guess.

[outer radish]

i guess

[ashen moth]

@torpid pine

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[ashen moth]

@outer radish @fading geyser @marble valve @swift portal The user still needs help with this help request.

[torpid pine]

sorry. had an exam yesterday

[torpid pine]

can you use the bot? it's a bit difficult to understand thru text

no wait.. I think I get what you mean.

is this correct?

[marble valve]

Well, I usually write integration by parts a bit differently, but yeah, that's correct.

[fading geyser]

Nice!

[outer radish]

good work

[torpid pine]

right. i actually like to solve problems like this otherwise i dont feel convinced... but, yes your original method of equating values with this formula (a^x f(x))' = a^x (f'(x) + f(x) ln(a)) was quicker and more efficient.

@outer radish @fading geyser @marble valve alright then. thank you all for helping me out. was stuck with this one for a while

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