Hey guys, I need some help on these questions:
Please help!
For 53: You want to perform long division first, because the degree of the numerator is greater than that of the denominator.
for 54: a) You need a partial fraction with denominator $\displaystyle (px+q)^k$ for each k value between 1 and m. The same applies to part b)
For 55:
a) Multiply the numerator by 2 and divide by 2 before the integral sign. Note that the new numerator is the derivative of the denominator, so the resulting antiderivative is a one half times the logarithm of the denominator... plus a constant, of course.
b) partial fractions ... factor denominator, then split into fractions.
c) substitute $\displaystyle u = (x+1)$
Good luck!
Hello, help1!
Long division: .$\displaystyle \frac{x^3}{x-5} \;=\;x^2 + 5x + 25 + \frac{125}{x-5}$53. What is the first step when integrating: .$\displaystyle \int\frac{x^3}{x-5}\,dx$
Then integrate: .$\displaystyle \int\left(x^2+5x+25 + \frac{125}{x-5}\right)\,dx$
. . $\displaystyle \frac{N(x)}{(px+q)^m} \;=\;\frac{A}{px+q} + \frac{B}{(px+q)^2} + \frac{C}{(px+q)^3} + \hdots \frac{N}{(px+q)^m} $54. Describe the decomposition of the rational function $\displaystyle \frac{N(x)}{D(x)}$
(a) if $\displaystyle D(x) \:=\:(px+q)^m$
$\displaystyle \frac{N(x)}{(ax^1+bx+c)^n}$(b) if $\displaystyle D(x) \:=\:(ax^2+bx+c)^m$, where $\displaystyle ax^2+bx+c$ is irreducible.
. . $\displaystyle =\;\frac{Ax+B}{ax^2+bx+c} + \frac{Cx+D}{(ax^2+bx+c)^2} + \frac{Ex+F}{(ax^2+bx+c)^3} + \hdots +\frac{Mx+N}{(ax^2+bx+c)^n}$