Reduction Formula Question (1 Viewer)

hscstudnet · Jul 24, 2020

I am struggling to do this question. I tried solving it and yeah thats pretty much it. I tried

.

Also generally, I can do integration by parts questions very well but whenever it comes to reduction formula or proof by Induction questions where there is a recursive formula involved, I only get it right 30-50% of the times. What are some clues or hints I should look for when proving the formula in both these types of questions.

Thanks in advance.

quickoats · Jul 24, 2020

The hint is in the “algebraic manipulation” part of the question. Is there anything you can do to the numerator to help you get the required result?

With reduction questions, you kinda just have to trial and error a solution through recognition. As you do more questions, you’ll be able to recognise bits (from integration by parts) that pop up every now and then.

hscstudnet · Jul 24, 2020

quickoats said:
The hint is in the “algebraic manipulation” part of the question. Is there anything you can do to the numerator to help you get the required result?

With reduction questions, you kinda just have to trial and error a solution through recognition. As you do more questions, you’ll be able to recognise bits (from integration by parts) that pop up every now and then.

Thank you so much I got the answer

CM_Tutor · Jul 25, 2020

My first approach would be to rearrange the integrand to facilitate integration - a change like $\sec^n x = \sec^2 x \sec^{n - 2} x$ . Having an $I_n$ and an $I_{n-2}$ in the formula gives a clue as to what to separate.

In this case, this would give:

$\begin{align*} I_n = \int \frac{x^n}{1 + x^2} \; dx &= \int \frac{x^{n-2}.x^2}{1 + x^2} \; dx \\ &= \int \frac{x^{n-2}(1 + x^2 - 1)}{1 + x^2} \; dx \\ &= \int x^{n-2}\left(\frac{1 + x^2 - 1}{1 + x^2}\right) \; dx \\ &= \int x^{n-2}\left(1 - \frac{1}{1 + x^2}\right) \; dx \\ &= \int x^{n-2} - \frac{x^{n-2}}{1 + x^2} \; dx \\ &= \frac{x^{n-2+1}}{n - 2 + 1} - \int \frac{x^{n-2}}{1 + x^2} \; dx \\ &= \frac{x^{n-1}}{n - 1} - I_{n-2} \end{align*}$

CM_Tutor · Jul 25, 2020

Another approach is to rearrange the target result, in this case by examining $I_n + I_{n-2}$ :

$\begin{align*} I_n + I_{n-2} &= \int \frac{x^n}{1 + x^2} \; dx + \int \frac{x^{n-2}}{1 + x^2} \; dx \\ &= \int \frac{x^n + x^{n-2}}{1 + x^2} \; dx \\ &= \int \frac{x^{n-2}(x^2 + 1)}{1 + x^2} \; dx \\ &= \int x^{n-2} \; dx \\ &= \frac{x^{n-2+1}}{n - 2 + 1} \\ &= \frac{x^{n-1}}{n - 1} \end{align*}$

This approach removes the need to determine what to separate (as in the above approach) but is difficult to apply if the coefficient of the term in $I_{n-1}$ or $I_{n-2}$ is not $\pm1$ .

hscstudnet · Jul 26, 2020

CM_Tutor said:
Another approach is to rearrange the target result, in this case by examining $I_n + I_{n-2}$ :

$\begin{align*} I_n + I_{n-2} &= \int \frac{x^n}{1 + x^2} \; dx + \int \frac{x^{n-2}}{1 + x^2} \; dx \\ &= \int \frac{x^n + x^{n-2}}{1 + x^2} \; dx \\ &= \int \frac{x^{n-2}(x^2 + 1)}{1 + x^2} \; dx \\ &= \int x^{n-2} \; dx \\ &= \frac{x^{n-2+1}}{n - 2 + 1} \\ &= \frac{x^{n-1}}{n - 1} \end{align*}$

This approach removes the need to determine what to separate (as in the above approach) but is difficult to apply if the coefficient of the term in $I_{n-1}$ or $I_{n-2}$ is not $\pm1$ .

Thanks for both the methods do you have any tips on proof questions also inequality proofs there are pretty annoying

CM_Tutor · Jul 26, 2020

hscstudnet said:
Thanks for both the methods do you have any tips on proof questions also inequality proofs there are pretty annoying

Probably best to ask specific questions / examples - I'll put in advice where I can.

po45gp]sedorgmjkpoerdjgf · Aug 8, 2020

hscstudnet said:
I am struggling to do this question. I tried solving it and yeah thats pretty much it. I tried .
View attachment 28593
Also generally, I can do integration by parts questions very well but whenever it comes to reduction formula or proof by Induction questions where there is a recursive formula involved, I only get it right 30-50% of the times. What are some clues or hints I should look for when proving the formula in both these types of questions.

Thanks in advance.

thats a rare type of reduction question. The vast majority use IBP or a trig identity.

Trebla · Aug 8, 2020

As a rule of thumb, if you see reduction formulae in the form:
$I_{n} + bI_{n-k} + c = 0$
and if b is independent of n then it is likely that a simple addition approach would work.

Integration by parts usually generates a recurrence of the original integral as well as a bunch of factors dependent on n (due to the nature of differentiating and integrating involved).

hscstudnet · Aug 9, 2020

Trebla said:
As a rule of thumb, if you see reduction formulae in the form:
$I_{n} + bI_{n-k} + c = 0$
and if b is independent of n then it is likely that a simple addition approach would work.

Integration by parts usually generates a recurrence of the original integral as well as a bunch of factors dependent on n (due to the nature of differentiating and integrating involved).

This is a such good way to identify the type, thank you so much.

Reduction Formula Question (1 Viewer)

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