2 Qs: int [x^2/sqrt(9 - x^2)] dx, int [x^3/(x^2-2x+1)] dx

[killasnake]

Hi I have a few questions, on how to solve these two intergral problems.

1) \(\displaystyle x^2/
sqrt(9-x^2) dx\)

With this problem I am suppose to use Rationalizing Substitution right?

\(\displaystyle sqrt(a^2+x^2)\) Substitution is x= a sin t

x= 3 sin t
dx = 3 cos t

\(\displaystyle sqrt(3^2-x^2) dx\) = 3 cos t

2) \(\displaystyle x^3/x^2-2x+1 dx\)

With this problem i am suppose to use proper rational function correct?

__A__ + __B__
(x-1)____(x-1)

Solve for a and b and then take the intergral correct?

 

[stapel]

1) What happened to the "x²" in the original numerator? What is your final answer? Did it differentiate back to the original integrand?

I would try integration by parts, noting that:

. . . . .v = (1/3) Sin^-1(x/3)

. . . . .dv = 1/sqrt[9 - x²]

2) First, do the long division, to get a "mixed number"; that is, simplify first to a polynomial plus a proper fraction:

. . . . .x + 2 + (3x - 2)/(x² - 2x + 1)

Then do the partial-fraction decomposition.

Eliz.