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Section 6.4

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1. (a) \displaystyle{\int \frac{x-1}{x} \ dx = x - \log|x| + c}

(c) \displaystyle{\int \frac{3x^2}{x-2} \ dx = \frac{3}{2}x^2 + 6x + 12\log|x-2| + c}

(e) \displaystyle{\int \frac{4x+1}{2x^2+x-3} \ dx = \log|2x^2 + x - 3| + c}

2. (a) \displaystyle{\int \frac{1}{(x+2)(x-4)} \ dx = \frac{1}{6}\log|x-4| - \frac{1}{6}\log|x+2| + c}

(c) \displaystyle{\int \frac{3x}{(2x+3)(x+1)} \ dx = \frac{9}{2}\log|2x+3| - 3\log|x+1| + c}

(e) \displaystyle{\int \frac{x}{x^2+x-6} \ dx = \frac{2}{5}\log|x-2| + \frac{3}{5}\log|x+3| + c}

(g) \displaystyle{\int \frac{3}{x^2+5x+6} \ dx = 3\log|x+2| - 3\log|x+3| + c}

3. (a) \displaystyle{\frac{1}{(x-1)^2} \ dx = -\frac{1}{x-1} + c}

(c) \displaystyle{\int \frac{x}{x^2+2x+1} \ dx = \frac{1}{x+1} + \log|x+1| + c}

(e) \displaystyle{\int \frac{5}{(x+2)^3} \ dx = -\frac{5}{2(x+2)^2} + c}

(g) \displaystyle{\int \frac{3x^2}{(x+1)^2(x-3)} \ dx = \frac{3}{4(x+1)} + \frac{21}{16}\log|x+1|}
\displaystyle{+ \frac{27}{16}\log|x-3| + c}

4. (a) \displaystyle{\int \frac{1}{(3x+2)^2} \ dx = -\frac{1}{3(3x+2)} + c}

(c) \displaystyle{\int \frac{9x^2-4x}{3x^3-2x^2+5} \ dx = \log|3x^3 - 2x^2 + 5| + c}

(e) \displaystyle{\int_{-1}^1 \frac{1}{x^2-4} \ dx = -\frac{1}{2}\log(3)}

(g) \displaystyle{\int \frac{4x+5}{(x-2)^2(x+5)} \ dx = -\frac{13}{7(x-2)} + \frac{15}{49}\log|x-2|}
\displaystyle{- \frac{15}{49}\log|x+5| + c}

5. \displaystyle{x(t) = \frac{1-e^{2t}}{1+e^{2t}}}

This entry was posted on 9 August 9 2007 at 10:32 am and is filed under Section 6.4. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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