Question

Need a little help with some substitution integration.

Answer 1

Rational integrals are really confusing me for example see problem below

Answer 2

post

Answer 3

So for this example, after I have picked x^2+5 as du do I rewrite my integral as...

Answer 4

\[\int\limits_{}^{}\frac{ 1 }{ (u)^2 }\]

or is it just

\[\int\limits_{}^{}(u)^2\]

Answer 5

Er i forgot the three in front, just pretend its there

Answer 6

The latter with the 3 in the front.

Answer 7

Ok, I dont understand why you can just get rid of the fraction like that though. I would understand if u was in the numerator, but it seems counter intuitive to me if its in the denominator....if that makes any sense.

Answer 8

you are not "getting rid" of fraction you are using substitution so they are in effect the same but just manageable

Answer 9

OK, so if it helps you to think of it like this...
You replace every instance of \(x^2 + 5 \) in your integrand with \(u\) and every instance of \(2x dx\) in your integrand with \(du\).

Answer 10

Also,

\[3\int\limits_{}^{}(u)^2 = 3[\frac{ 1 }{ 3 }(u)^3] = 3[\frac{ (x^2+5)^3 }{ 3 }] = (x^2+5)^3\]

Answer 11

Did i do something incorrectly?

Answer 12

Yes, we said 1/u^2, not u^2

Answer 13

I think I understand what your saying @ParthKohli

Answer 14

oh ok.

Answer 15

does that mean I have to integrate again?

Answer 16

oh or ((u)^2)^-1

Answer 17

let me ask you, why did you say du = x^2+5

Answer 18

du=2x

Answer 19

did i say that?

Answer 20

\(u = x^2+5 \)
\(du = 2x ~dx \)

Answer 21

yea for sure

Answer 22

so what is next?

Answer 23

I need to rewrite it and solve it properly, but \[3\int\limits_{}^{}\frac{ 1 }{ (u)^2 } = 3\int\limits_{}^{} (u)^{-2}\]

Answer 24

so my answer should be...

Answer 25

\[-\frac{ 3 }{ (x^2+5) }\]

I think

Answer 26

+C

Answer 27

ye

Answer 28

Awesome, thank you!