Use Taylor polynomial to evaluate integral - II

[thanhle4110]

Find the Taylor series about x = 0 for this indefinite integral


The derivative got so long and time-consuming. Is there any other way faster than differentiating the function until the 4th, 5th order?

 

[Cubist]

Is there any other way faster than differentiating the function until the 4th, 5th order?

You could have posted some of the derivatives that you obtained ! And I'll assume that you don't want to use a computer algebra system to help you do this.

Did you notice that you always obtain a derivative of the form [math]P(x)\cdot e^{-x^3}[/math] where [math]P(x)[/math] is a polynomial in x?

Try differentiating when [math]P(x)=c_1x^{c_2}[/math] since this might help you to spot a pattern. Then you could tabulate the coefficients of the polynomials obtained for successive derivatives...

	x0	x1	x2	x3	x4	x5	etc...
f		1
f'	1			-3
f''			?			?
f'''		?			?
f''''	?			?			?
etc

When you've filled out a reasonable size table then you could try to determine if there is a direct expression for the x⁰ column entries (since you are doing the Taylor series around x=0 the higher coefficients x¹,x²,etc are not actually required).

 

[Cubist]

You could have posted some of the derivatives that you obtained !

I just re-read my post - and wanted to let you know that I didn't intend the exclamation mark as shouting but I wanted to express the importance of sharing your work since it enables tutors to help you more effectively. Anyway, good luck with progressing this and please post again if you're stuck (include a photo of your work if that is easier than typing).