4u reduction formula help (1 Viewer)

astj · Apr 23, 2024

hi I tried doing by parts for these questions but I can't seem to get the results, thank in advance.

liamkk112 · Apr 23, 2024

astj said:
hi I tried doing by parts for these questions but I can't seem to get the results, thank in advance.

for first one we can write x^n=x^(n-2) (1+x^2) -x^(n-2)
so the integral then becomes integral of x^n-2 - I_(n-2)

and then doing power rule gives u the result

Susu123 · Apr 23, 2024

liamkk112 said:
for first one we can write x^n=x^(n-2) (1+x^2) -x^(n-2)
so the integral then becomes integral of x^n-2 - I_(n-2)

and then doing power rule gives u the result

how do you think of manipulating the question like that during an exam?? or does it just come from practice

liamkk112 · Apr 23, 2024

Susu123 said:
how do you think of manipulating the question like that during an exam?? or does it just come from practice

practice ig, i just look at it and think "how do i get (x^2+1) in the numerator so i can cancel out the denominator"

Susu123 · Apr 23, 2024

liamkk112 said:
practice ig, i just look at it and think "how do i get (x^2+1) in the numerator so i can cancel out the denominator"

oh right, that makes sense, thanks!

ivanradoszyce · Apr 25, 2024

This looks very similar to the following problem:

$\begin{align*} \text{Given} \\ I_n &= \int \frac{x^n}{1 + x^2} dx \\ \text{show} \\ I_n &= \dfrac{x^{n - 1}}{n - 1} - I_{n - 2} \\ \text{From this expression $I_{n-2} = \int \frac{x^{n-2}}{1 + x^2} dx$ is required. }\\\\ \text{We can get this by adding and subtracting $ x^{n-2} $ .Thus }\\\\ \int \frac{x^n}{1 + x^2} dx &= \int \frac{x^n + x^{n-2} - x^{n-2}}{1 + x^2} dx \\ &= \int \frac{x^n + x^{n-2} }{1 + x^2} - \frac{x^{n-2}}{1 + x^2} dx \\ \text{We have established the last fraction as $I_{n-2}$ }\\\\ \text{This is the tricky bit - factor $x^{n - 2}$ from the numerator.}\\ &= \int \frac{x^{n - 2}(x^2 + 1) }{1 + x^2} - \frac{x^{n-2}}{1 + x^2} dx \\ &= \int x^{n - 2}dx - \int\frac{x^{n-2}}{1 + x^2} dx \\ &= \dfrac{x^{n - 1}}{n - 1} - I_{n - 2} \\ \end{align*}$

4u reduction formula help (1 Viewer)

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