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Math Help - Substitution questions

  1. #1
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    Substitution questions

    These are some questions on a practice quiz that I can't get, a little help please.

    \int \frac{b}{x^2+a^2} dx

    \int \sec(3x)\tan(3x)dx

    \int \frac{2x+13}{x^2+6x+15} dx

    \int \frac{8}{x^2-9} dx

    Also this one:
    "Fill in for the start of a Partial Fraction Decomposition - do not actually solve"
    \frac{some-quadratic}{x^3+x^2+3x+3}

    Thanks a lot
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  2. #2
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    Quote Originally Posted by nirva View Post
    These are some questions on a practice quiz that I can't get, a little help please.

    \int \frac{b}{x^2+a^2} dx
    If a\not = 0 Divide by it,
    \frac{b}{a^2}\int \frac{1}{1+x^2/a^2}dx
    For simplicify sake say a>0
    Then,
    \frac{b}{a^2}\int \frac{1}{1+(x/a)^2}dx
    The substitution function u=x/a will fast transform this into an inverse tangent.
    Thus,
    \frac{b}{a}\tan^{-1}(x/a)+C

    \int \frac{8}{x^2-9} dx
    Fractional decomposition,
    \frac{8}{6}\int \frac{1}{x-3}-\frac{1}{x+3}dx
    Thus,
    \frac{4}{3}\ln |x-3|-\frac{4}{3}\ln |x+3|+C=\frac{4}{3}\ln \left| \frac{x-3}{x+3} \right|+C
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  3. #3
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    Hello, nirva!

    Fill in for the start of a Partial Fraction Decomposition - do not actually solve

    . . . \frac{\text{some quadratic}}{x^3+x^2+3x+3}

    The denominator factors:
    . . x^3 + x^2 + 3x + 3 \;=\;x^2(x+1) + 3(x+1) \;=\;(x+1)(x^2+3)


    So we have: . \frac{\text{some quadratic}}{(x+1)(x^2+3)} \;= \;\frac{A}{x+1} + \frac{Bx + C}{x^2 + 3}

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