Integration: int [(3 x^(1/2) - x] dx

[jlmills5]

I am not understanding how to integrate [(3x^(1/2)-x]. I've attached a picture of this part of the problem (worked out). If anyone could explain to me step 5 to 6 of the picture I attached or I'd appreciate it.

Oh and also the points of intersection. I understand they are when x = 0, 3 from just looking at sq.rt (3x) = x [number 2 in the picture] but how is it solved for the x's?

 

[arthur ohlsten]

Re: Integration problem

int (3x)^1/2 dx
we need a 3 dx
multiply by 1/3 and 3
int (3x)^1/2dx = 1/3 int (3x)^1/2 [3dx]
int (3x)^1/2 dx =1/3 { [(3x)^3/2]/3/2}
int (3x^1/2) dx = [2/9] 3x^3/2

if 3x=z then
3 dx = dz or
dx=dz/3 this is the substitution you must make

Arthur

 

[Deleted member 4993]

Re: Integration problem

This is straight forward integration:

\(\displaystyle \int(ax)^n \, dx \, = \, a^n\cdot\int x^n \, dx \, = \, a^n\cdot\frac{1}{n+1}\cdot x^{n+1} \, + \, C\)

also points of intersections come from simple factorization:

\(\displaystyle \sqrt{3x} \, = \, x\)

\(\displaystyle x \, - \, \sqrt{3x} \, = \, 0\)

\(\displaystyle \sqrt x\cdot(\sqrt x \, - \, \sqrt{3}) \, = \, 0\)

Now do you see it....