# Integration by partial fractions: HELP!

• Jun 9th 2009, 03:49 PM
help1
Integration by partial fractions: HELP!
Hey guys, I need some help on these questions:

http://i43.tinypic.com/63wyvc.jpg

• Jun 9th 2009, 04:49 PM
Kaitosan
#55

(a.) Partial fractions
(b.) Partial fractions
(c.) Trigonometric substitution (use the "completing the square" operation)

As for #53, use polynomial division.

I don't understand #54 at all.
• Jun 9th 2009, 04:53 PM
apcalculus
Quote:

Originally Posted by help1
Hey guys, I need some help on these questions:

http://i43.tinypic.com/63wyvc.jpg

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!
• Jun 9th 2009, 06:15 PM
Soroban
Hello, help1!

Quote:

53. What is the first step when integrating: .$\displaystyle \int\frac{x^3}{x-5}\,dx$
Long division: .$\displaystyle \frac{x^3}{x-5} \;=\;x^2 + 5x + 25 + \frac{125}{x-5}$

Then integrate: .$\displaystyle \int\left(x^2+5x+25 + \frac{125}{x-5}\right)\,dx$

Quote:

54. Describe the decomposition of the rational function $\displaystyle \frac{N(x)}{D(x)}$

(a) if $\displaystyle D(x) \:=\:(px+q)^m$

. . $\displaystyle \frac{N(x)}{(px+q)^m} \;=\;\frac{A}{px+q} + \frac{B}{(px+q)^2} + \frac{C}{(px+q)^3} + \hdots \frac{N}{(px+q)^m}$

Quote:

(b) if $\displaystyle D(x) \:=\:(ax^2+bx+c)^m$, where $\displaystyle ax^2+bx+c$ is irreducible.
$\displaystyle \frac{N(x)}{(ax^1+bx+c)^n}$

. . $\displaystyle =\;\frac{Ax+B}{ax^2+bx+c} + \frac{Cx+D}{(ax^2+bx+c)^2} + \frac{Ex+F}{(ax^2+bx+c)^3} + \hdots +\frac{Mx+N}{(ax^2+bx+c)^n}$