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Math Help - Integration by partial fractions: HELP!

  1. #1
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    Integration by partial fractions: HELP!

    Hey guys, I need some help on these questions:



    Please help!
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  2. #2
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    #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.
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  3. #3
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by help1 View Post
    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 (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 u = (x+1)

    Good luck!
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  4. #4
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    Hello, help1!

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

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



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

    (a) if D(x) \:=\:(px+q)^m
    . . \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}



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

    . . =\;\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}

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