Why is integration by substitution used here..

When Integrating this equation, why is substitution used and not integrating by parts?

\int \:xe^{x^2}

And how can I identify when to use integrating by parts or Integration by substitution?

Reply 1

RDKGames

Study Forum Helper

Original post by joyoustele

When Integrating this equation, why is substitution used and not integrating by parts?

\int \:xe^{x^2}

And how can I identify when to use integrating by parts or Integration by substitution?

You won’t get anywhere with IBP (try it) and also upon inspection, we see that if we sub for x^2 then the derivative will be linear which would cancel the x next to e

Reply 2

joyoustele

Original post by RDKGames

You won’t get anywhere with IBP (try it) and also upon inspection, we see that if we sub for x^2 then the derivative will be linear which would cancel the x next to e

Okay, so in cases such as these step back and see which would be easiest to use, substitution or IBP.

Thanks :smile:

Reply 3

RDKGames

Study Forum Helper

Original post by joyoustele

Okay, so in cases such as these step back and see which would be easiest to use, substitution or IBP.

Thanks :smile:

Yes. Without actually doing the IBP, you can tell it wont work because you can choose u=x and v’=e^x^2 but you do not know how to integrate v’. Switching the order, we can diff e^x^2 but we’ll keep integrating x to higher powers so we’ll only make the problem more complicated

Reply 4

Gregorius

Original post by joyoustele

When Integrating this equation, why is substitution used and not integrating by parts?

\int \:xe^{x^2}

And how can I identify when to use integrating by parts or Integration by substitution?

I'm puzzled: why would you use a substitution? If you think about what

e^{x^2}

differentiates to, you can see that this is integragble by inspection.

Reply 5

joyoustele

Original post by Gregorius

I'm puzzled: why would you use a substitution? If you think about what

e^{x^2}

differentiates to, you can see that this is integragble by inspection.

I dont understand, Im sure I cannot just Isolate

e^{x^2}

on it's own when it is multiplied by an

x

(edited 6 years ago)

Reply 6

RDKGames

Study Forum Helper

Original post by joyoustele

I dont understand, Im sure I cannot just Isolate

e^{x^2}

on it's own when it is multiplied by an

x

You misunderstand. What he's saying is that if you look at the derivative of

e^{x^2}

you should notice that it's very similar to the expression you need to integrate, so just do a small change and you find your answer.

Reply 7

Notnek

Original post by joyoustele

I dont understand, Im sure I cannot just Isolate

e^{x^2}

on it's own when it is multiplied by an

x

Consider the derivative of

e^{x^2}

then work backwards.

Do you use inspection / pattern recognition when integrating? You can always use a substitution if you're not used to this kind of thinking, as long as you know what needs to be substituted.

Reply 8

K-Man_PhysCheM

Original post by joyoustele

I dont understand, Im sure I cannot just Isolate

e^{x^2}

on it's own when it is multiplied by an

x

What is the derivative of

e^{x^2}

?

Hence, do you see how you can integrate your expression by inspection?

Reply 9

joyoustele

Original post by RDKGames

You misunderstand. What he's saying is that if you look at the derivative of

e^{x^2}

you should notice that it's very similar to the expression you need to integrate, so just do a small change and you find your answer.

hmm Okay

Reply 10

thekidwhogames

One way to notice whether to use IBP or U-sub:

Notice if you consider any of the functions in the composite function, say f(x). If f'(x) cancels any other term - use U sub.

In this case

Notice if f(x)=x^2 then f'(x) = 2x which cancels nicely with the x term.

Reply 11

Hasufel

Original post by joyoustele

When Integrating this equation, why is substitution used and not integrating by parts?

\int \:xe^{x^2}dx

And how can I identify when to use integrating by parts or Integration by substitution?

This is just

\frac{1}{2}\int \:f ' (x)e^{f(x)}dx

you can use ibp if you want. (what's the derivative of e^(f(x))?)
Personally I'd use tabular integration, but I don't suppose school allow that kind of "cheating".