I hope that regular readers will not feel shortchanged if I post math questions here. The few times I’ve brought up a math question, it stimulated many commenters with excellent explanations and good discussion among those who like this stuff.
That said, here’s my issue. I’m getting ready to take Calc 2 this fall, and I find that the algebra is still the hardest thing. I understand the calculus concepts just fine; it’s untangling the algebra that gets me, and I suspect gets most math students.
⌠ x/(x+1)^1/2 dx = ⌠ u-1/√u du Subsitute u = x+1, du = dx
= ⌠ (√u – 1/√u) du Rewrite integrand
– ⌠ (u^1/2 – u^-1/2) du Fractional powers
integrate each individually — got all this OK
⌠ 2/3u^3/2 -2u^1/2 +C Antiderivatives
2/3(x+1)^3/2 – 2(x+1)^1/2 + C Replace U by X=1 (so far so good, but it’s the next thing that I don’t understand)
2/3(x+1)^1/2(x-2)+C Factor out (x+1)^1/2 and simplify
For the life of me, I don’t see how factoring out x+1)^1/2 leaves you with 2/3(x+1)^1/2(x-2) Wouldn’t factoring out 1/2 power from 3/2 power leave you with a power of 3? Of course you have a power of 1 on (x-2). I see a negative sign and a 2 there, but how did the x get in front of it to make x-2? How did (x+1)^3/2 become (x+1)^1/2?
Thanks in advance to the commenters. The book I’m using is Briggs and Cochran, Calculus Early Transcendentals, page 261, and the Student Solutions manual. This is the best book and SS manual that I’ve found.