# Reduction Formula Question (1 Viewer)

#### hscstudnet

##### Member
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.

#### quickoats

##### Well-Known Member
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

##### Member
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

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

In this case, this would give:

\bg_white \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

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

\bg_white \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 $\bg_white I_{n-1}$ or $\bg_white I_{n-2}$ is not $\bg_white \pm1$.

#### hscstudnet

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

\bg_white \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 $\bg_white I_{n-1}$ or $\bg_white I_{n-2}$ is not $\bg_white \pm1$.
Thanks for both the methods do you have any tips on proof questions also inequality proofs there are pretty annoying

#### CM_Tutor

##### Moderator
Moderator
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

##### New Member
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.

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

#### Trebla

As a rule of thumb, if you see reduction formulae in the form:
$\bg_white I_{n} + bI_{n-k} + c = 0$
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

##### Member
As a rule of thumb, if you see reduction formulae in the form:
$\bg_white I_{n} + bI_{n-k} + c = 0$
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.