Solving by substitution (1 Viewer)

Brontecat · Oct 11, 2009

I'm having difficulty with this q & don't know where to go at the end of the last step below:
Thanks

$\int x^2 \sqrt{x^3+1} || \; Let\:u=x^3 +1 || \; \frac{du}{dx}= 3x^2 || \; \frac{du}{3x^2}= dx || \; x^2 \int \sqrt{u}\; \frac{du}{3x^2}||$

Trebla · Oct 11, 2009

First of all, you CANNOT take the variable out of the integration operator. The x² term must stay inside the integral!!!

$u = x^3+1\\du=3x^2\,dx\\\\\int x^2\sqrt{x^3+1}\,dx=\dfrac{1}{3}\int 3x^2\sqrt{x^3+1}\,dx\\\\= \dfrac{1}{3}\int \sqrt{u}\,du\\\\=\dfrac{2}{9}u^{\frac{3}{2}}+c\\\\= \dfrac{2}{9}(x^3+1)^{\frac{3}{2}}+c$

lychnobity · Oct 11, 2009

$\int x^{2}\sqrt{u}\frac{du}{3x^{2}}\\ = \int \frac{\sqrt{u}}{3} du\\ = \frac{1}{3}\times \frac{2}{3}\times u\sqrt{u} + C\\ = \frac{2u\sqrt{u}}{9} + C\\ = \frac{2(x^{3}+1)\sqrt{x^{3}+1}}{9} + C$

Edit: beaten by Trebla.

Also note that $u\sqrt{u}$ is $u^\frac{3}{2}$

hermand · Oct 11, 2009

Trebla said:
First of all, you CANNOT take the variable out of the integration operator. The x² term must stay inside the integral!!!

$u = x^3+1\\du=3x^2\,dx\\\\\int x^2\sqrt{x^3+1}\,dx=\dfrac{1}{3}\int 3x^2\sqrt{x^3+1}\,dx\\\\= \dfrac{1}{3}\int \sqrt{u}\,du\\\\=\dfrac{2}{9}u^{\frac{3}{2}}+c\\\\= \dfrac{2}{9}(x^3+1)^{\frac{3}{2}}+c$

this.

it's a lot simpler and easier.

Brontecat · Oct 11, 2009

Trebla said:
First of all, you CANNOT take the variable out of the integration operator. The x² term must stay inside the integral!!!

woops, i thought because it was a spectator that was ok

Thanks for all the replies, i'm unstuck now

4028023 said:
wow Brontecat ur hsc is in 2010 but your doing the maths and bio hsc this yr! how did you get to do that?

and out of curiosity what school do you go to?

At school instead of doing 5 periods of biology, like most school's do I do 10 periods and put in extra effort at home to get the work done. It's kind of like accelerating, but everyone at my school does it.
The idea behind it is that you can do less HSC exams all at once. So next year, i'll do year 11 and 12 adv. english in the one year.

ninetypercent · Oct 11, 2009

but do u do the hsc exam for mathematics and biology this year?

Brontecat · Oct 12, 2009

yep, sure do

Brontecat · Oct 13, 2009

I seriously had no idea this was 3 unit (i'm taking your word for it)
It was taught to me at school (im my 2U maths class) i just assumed that it was 2U content - i am definitely not trying to show off, just asking for some assistance in stuff i didn't understand

Solving by substitution (1 Viewer)

Brontecat

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Trebla

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lychnobity

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hermand

je t'aime.

Brontecat

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ninetypercent

ninety ninety ninety

Brontecat

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Brontecat

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