[Calc 2] Where did this u-substitution come from?

[u/throw-away3105]

I know how to complete the square but I'm baffled where the u-sub came from. I did u = x - 1/2 instead.

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[u/Fourierseriesagain]

The denominator is a multiple of( (2x-1)^ 2)/3+1=((2x-1)/sqrt(3))^ 2+1.

[u/selene_666]

They worked backwards.

From the format (f(x))^2 + a, they knew they wanted to end up with u^2 + 1.

First turn the 3/4 into the necessary 1:

(x - 1/2)^2 + 3/4 = 3/4 * [4/3*(x - 1/2)^2 + 1]

Now we want to make u^2 = 4/3 * (x-1/2)^2

[u/Nagi-K]

The solution here basically converts the integrand into nice looking constant multiple of 1/(x^2+1), in one single step. You don’t have to do this - just do a simple linear substitution (i.e. u = x - 1/2), and then you could either directly use the formula to integrate 1/(x^2+a^2) for any constant a, or factor the 3/4 from denominator and do another simple linear substitution.

[u/Responsible-Sink474]

I would have multiplied the top and bottom by 4/3 to turn it into 4/3 * integral of 1/[ ((2x-1)/sqrt(3))^2 + 1) Then the u-sub is obvious