1. ## Integration by substitution

Hey,

I have this question:

$\displaystyle \int \frac{x}{\sqrt{3-x}}$

My book says to use the substition $\displaystyle u^2 = 3-x$

Using that substition i get a du that is some utter rubbish and couldn't help in solving this.

If i use u = 3-x instead i get an answer but it's still wildly different from the books.

How could i use the u^2 substitution?

Thanks

2. The integrand = -2x * 'rubbish'

3. This is what i get - my 'rubbish' so to speak? Is my approach to the differentiation wrong?

$\displaystyle u^2 = 3-x$

$\displaystyle du = -\frac{1}{2}(3-x)^{\frac{3}{2}}$

4. You took the power up a unit - down instead! i.e. minus 1/2 not plus 3/2

5. ARGHH - i've really got to stop making this rookie mistakes!! Very sorry!

so...

$\displaystyle du = -\frac{1}{2}(3-x)^{-\frac{1}{2}}$

becomes...

$\displaystyle du = - \frac{1}{2\sqrt{3-x}}$

And finally i get:

$\displaystyle dx = (-2\sqrt{3-x})du$

If thats right, how does that go back into the function. usually i have just a constant to take outside the integral. Sorry for all the questions.

Thanks

6. No prob. Again, all you need do is express -2x in terms of u then integrate with respect to u. I suppose someone may fill in the whole story. I'll add a pic in a couple of mins.

Here we are. Just in case a picture helps...

i.e...

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Spoiler:

__________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

7. Thanks,

My book has the answer as $\displaystyle -\frac{2}{3}(x+6)\sqrt{(3-x)}$

Does this match your answer at all - I can't quite understand the balloon pictures but i appreciate the effert. I've made a note to learn the balloons after im done with integration.

Thanks

8. You'll find this place integrate x/sqrt(3 - x) - Wolfram|Alpha useful and if you click on 'show steps' you'll see that yes the two answers are equivalent. (Interestingly, they haven't used quite the same u.)

Oh gosh don't worry about my crazy method! Whatever works for you.

9. What an amazing website! Didn't know things like that existed. That last step is beyond me but im going to go through it now and MAKE it make sense.

Thanks for all your help - hope i didn't waste your time too much