Integrating the square root of a linear function.

[cptrsn]

Homework Statement

$$\int\sqrt{2*x-1}$$

Homework Equations

The Attempt at a Solution

Homework Statement

Obviously this can be fixed with the antiderivative of a linear function with proper constants i.e. $$\int A\times f(bx + c) = \frac{A}{b} F(bx + c)$$, however my instructor provided me with a "clue" which I somehow can't seem to work out.

He claims that $$\int\sqrt{2*x-1}$$ is equivilant to $$\int u^2$$ using "the right" substitution. My idea usually is:
1) $$u = \sqrt{2*x - 1}$$
2) $$x = u^2$$
3) $$dx = 2u du$$

Which would lead to $$\int u dx = \int u 2u du = \int 2u^2 du$$.
Obviously this is NOT identical to what he suggests, so can anyone point me in the right direction? I'm quite keen to know how he see such a problem.

 

[rootX]

If u = sqrt(2x-1)
then how x = u^2?

 

[Bohrok]

I don't think he quite knew what he was talking about; there are some problems with that which I wouldn't even try to work with or fix.
What part of the integrand can u be that will easily let you integrate with respect to u?

 

[HallsofIvy]

The teacher not being here to defend himself, I might suggest that cptrsn did not quite remember what the teacher had suggested. Any time you have a linear term, ax+ b, an obvious substitution is just u= ax+ b so that du= adx.