Question

what is the integral from 0 to infinity of (x^3)/((x^8)+1) help plzzzz

Answer 1

Note that \(\large\displaystyle \int_0^{\infty} \frac{x^3}{x^8+1}\,dx = \int_0^{\infty}\frac{x^3}{(x^4)^2+1}\,dx\). Now make the substitution \(\large u =x^4\).

Can you take things from here? :-)

Answer 2

but the numerator will u^-1 ?

Answer 3

No, it will not. If \(\large \color{blue}{u=x^4}\), then \(\large \,du =4x^3\,dx\implies \color{red}{\dfrac{du}{4}=x^3\,dx}\). Thus,
\[\large \int_0^{\infty}\frac{\color{red}{x^3}}{(\color{blue}{x^4})^2+1}\color{red}{\,dx} \xrightarrow{u=x^4}{}\int_0^{\infty} \frac{\color{red}{1}}{\color{red}{4}(\color{blue}{u}^2+1)}\color{red}{\,du} = \frac{1}{4}\int_0^{\infty}\frac{1}{u^2+1}\,du\]
Does this clarify things? Do you think you can take things from here? :-)

Answer 4

yeah i thought about that when i differentiate but when will subtitute back to x^4 ?

Answer 5

after i take the limit

Answer 6

You can change your limits of integration so you don't have to back substitute after finding the integral. In this case, when you change the limits of integration, you get the same limits back.

Answer 7

is it Arctan of U around 0 and infinity

Answer 8

In particular, if \(x=0\), then \(u(0) = 0^4 = 0\); thus the lower limit for the variable u is 0. Likewise, as \(x\to infty\), we see that \(\displaystyle \lim_{x\to\infty} u(x) = \lim_{x\to\infty} x^4 = \infty \); hence the upper limit for the variable u is \(\infty\).

And yes, \(\large\displaystyle \int_0^{\infty} \frac{1}{u^2+1}\,du = \left.\bigg[\arctan u\right]\bigg|_{0}^{\infty}\).

Answer 9

whats my new limit of integration plzz

Answer 10

nice so it wont change

Answer 11

Exactly; that's the nice thing about changing the limits of integration in this case.

Answer 12

However, don't forget the factor of \(\large \dfrac{1}{4}\) in your integral!

Answer 13

you are the besssssst

Answer 14

I just want to make sure you got the right answer; what did you get as the final answer? :-)

Answer 15

yesir its 1/4 times pi/2

Answer 16

pi/8

Answer 17

Exactly, which is \(\large\dfrac{\pi}{8}\). I hope this made sense! :-)

Answer 18

yes it did u are truly my savior