Hi I've been given this question and I'm not sure what to do with u^2. Wouldn't you just make u=3+x or is there a reason that it's u^2?
#u^2 Substitution Integration
real crystal
sterile valley
Hi I've been given this question and I'm not sure what to do with u^2. Wouldn't ...
Well, we have an expression under the root and we want to get rid of the root.
With thus substitution we have:
u^2 = 3 + x, x = u^2 - 3, dx = 2udu
So:
2x√(3 + x)dx = 2(u^2 - 3)u (2udu) = 4u^2 (u^2 - 3)du
This is now just a polynomial, which is easy to integrate.
Later, you will learn a more general kind of substitution to deal with roots of linear expressions or of ratios of linear expressions.
real crystal
Ok thank you
One more thing..
When you differentiate the u^2 = 3+x
You get d^2u/dx = 1 right?
How does that become 2udu
sterile valley
No, you need to take the differential of both sides.
x = u^2 - 3
dx = d(u^2 - 3)
3 is a constant, so its differential is 0.
dx = d(u^2)
u^2 is a power function, for which we have d(u^2)/du = 2u. So, d(u^2) = 2udu.
dx = 2udu