finding indefinite integral ∫(x²/(x-1)) dx?

You can integrate ∫ x^2/(x - 1) dx by adding and subtracting 1 in the numerator to get:

∫ x^2/(x - 1) dx

= ∫ [(x^2 - 1) + 1]/(x - 1) dx

= ∫ [(x^2 - 1)/(x - 1) + 1/(x - 1)] dx

= ∫ [(x + 1)(x - 1)/(x - 1) + 1/(x - 1)] dx, via difference of squares

= ∫ [(x + 1) + 1/(x - 1)] dx.

(You could of also divided, but this is an easier approach)

Then, integrating term-by-term gives:

∫ [(x + 1) + 1/(x - 1)] dx

= ∫ x dx + ∫ dx + ∫ 1/(x - 1) dx

= (1/2)x^2 + x + ln|x - 1| + C.

I hope this helps!

right this is a fashion no longer regarding trig: (as Brian wisely talked approximately, i assume you meant a million /?(x² + 4)³ somewhat than a million /?(x² + 4)^(3/2)..) ? [a million /?(x² + 4)³] dx = divide and multiply by potential of four (with the intention to get 4 on the precise): (a million/4) ? [4 /?(x² + 4)³] dx = then upload and subtract x² on the precise: (a million/4) ? {[(x² + 4) - x²] /?(x² + 4)³} dx = wreck it up into: (a million/4) ? [(x² + 4) /?(x² + 4)³] dx - (a million/4) ? [x² /?(x² + 4)³] dx = the 1st integrand simplifies into: (a million/4) ? [a million /?(x² + 4)] dx - (a million/4) ? [x² /?(x² + 4)³] dx (#) enable's now combine the 2d critical by potential of aspects, rewriting it as: ? x [x /?(x² + 4)³] dx letting: x = u ? dx = du [x /?(x² + 4)³] dx = dv (dividing and multiplying by potential of two to get the by-fabricated from (x² + 4) on the precise) (a million/2) [2x dx /?(x² + 4)³] = dv (a million/2) [d(x² + 4) /?(x² + 4)³] = dv (a million/2) (x² + 4)^(-3/2) d(x² + 4) = dv (a million/2) {a million/[(-3/2)+a million]} (x² + 4)^[(-3/2)+a million] = v (a million/2)[a million/(-a million/2)](x² + 4)^(-a million/2) = v (a million/2)(- 2) [a million /?(x² + 4)] = v [- a million /?(x² + 4)] = v yielding: ? u dv = u v - ? v du ? x [x /?(x² + 4)³] dx = x [- a million /?(x² + 4)] - ? [- a million /?(x² + 4)] dx ? [x² /?(x² + 4)³] dx = [- x /?(x² + 4)] + ? [a million /?(x² + 4)] dx enable's now plug this into the above (#) expression: (a million/4) ? [a million /?(x² + 4)] dx - (a million/4) ? [x² /?(x² + 4)³] dx = (a million/4) ? [a million /?(x² + 4)] dx - (a million/4) {[- x /?(x² + 4)] + ? [a million /?(x² + 4)] dx} = (a million/4) ? [a million /?(x² + 4)] dx + (a million/4)[x /?(x² + 4)] - (a million/4) ? [a million /?(x² + 4)] dx = for that reason, opposite words canceling out, we finally end up with: (a million/4)[x /?(x² + 4)] + C so the respond is: ? [a million /?(x² + 4)³] dx = {x /[4?(x² + 4)]} + C i'm hoping this facilitates

-(3/2)+x+x^2/2+Log[-1+x]