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webwork / ma114f25 / a4-partial fractions / 6
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a4-partial fractions: problem 6
(2 points)
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consider the indefinite integral \\(\int\frac{5x^{3}+8x^{2}+75x + 150}{x^{4}+25x^{2}}dx\\). then the integrand has partial fractions decomposition
where
\\(a =\\)
\\(b =\\)
\\(c =\\)
\\(d =\\)
integrating term by term, we obtain that \\(\int\frac{5x^{3}+8x^{2}+75x + 150}{x^{4}+25x^{2}}dx=)
note: you can earn full credit by answering just the last part.
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Explanation:

Step1: Factor the denominator

Factor $x^{4}+25x^{2}=x^{2}(x^{2} + 25)$. Then we assume the partial - fraction decomposition of $\frac{5x^{3}+8x^{2}+75x + 150}{x^{4}+25x^{2}}$ is of the form $\frac{5x^{3}+8x^{2}+75x + 150}{x^{2}(x^{2}+25)}=\frac{a}{x}+\frac{b}{x^{2}}+\frac{cx + d}{x^{2}+25}$.
Cross - multiply: $5x^{3}+8x^{2}+75x + 150=a x(x^{2}+25)+b(x^{2}+25)+(cx + d)x^{2}$.

Step2: Find the value of b

Let $x = 0$. Then $5(0)^{3}+8(0)^{2}+75(0)+150=b(0 + 25)$, so $b = 6$.

Step3: Expand and group terms

Expand the right - hand side: $a x(x^{2}+25)+b(x^{2}+25)+(cx + d)x^{2}=ax^{3}+25ax+bx^{2}+25b+cx^{3}+dx^{2}=(a + c)x^{3}+(b + d)x^{2}+25ax+25b$.
Comparing the coefficients of the like - power terms:
For the coefficient of $x^{3}$: $5=a + c$.
For the coefficient of $x^{2}$: $8=b + d$. Since $b = 6$, then $d=2$.
For the coefficient of $x$: $75 = 25a$, so $a = 3$.
Since $a = 3$ and $5=a + c$, then $c=2$.

Step4: Integrate term - by - term

$\int\frac{5x^{3}+8x^{2}+75x + 150}{x^{4}+25x^{2}}dx=\int\frac{3}{x}dx+\int\frac{6}{x^{2}}dx+\int\frac{2x+2}{x^{2}+25}dx$.
$\int\frac{3}{x}dx=3\ln|x|$.
$\int\frac{6}{x^{2}}dx=6\int x^{-2}dx=- \frac{6}{x}$.
$\int\frac{2x+2}{x^{2}+25}dx=\int\frac{2x}{x^{2}+25}dx+\int\frac{2}{x^{2}+25}dx$.
$\int\frac{2x}{x^{2}+25}dx=\ln(x^{2}+25)$ and $\int\frac{2}{x^{2}+25}dx=\frac{2}{5}\arctan(\frac{x}{5})$.

So $\int\frac{5x^{3}+8x^{2}+75x + 150}{x^{4}+25x^{2}}dx=3\ln|x|-\frac{6}{x}+\ln(x^{2}+25)+\frac{2}{5}\arctan(\frac{x}{5})+C$.

Answer:

$a = 3$
$b = 6$
$c = 2$
$d = 2$
$\int\frac{5x^{3}+8x^{2}+75x + 150}{x^{4}+25x^{2}}dx=3\ln|x|-\frac{6}{x}+\ln(x^{2}+25)+\frac{2}{5}\arctan(\frac{x}{5})+C$