QuestionJune 6, 2025

Evaluate the integral. int _(6)^8(x)/(x^2)+4x+20dx square

Evaluate the integral. int _(6)^8(x)/(x^2)+4x+20dx square
Evaluate the integral. int _(6)^8(x)/(x^2)+4x+20dx square

Solution
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Answer

\frac{1}{2} \ln \frac{33}{23} Explanation 1. Simplify the Denominator The denominator x^2 + 4x + 20 can be rewritten as (x+2)^2 + 16 by completing the square. 2. Use Substitution Let u = x^2 + 4x + 20, then du = (2x + 4)dx. Rewrite x dx in terms of du: x dx = \frac{1}{2}(du - 4dx). 3. Adjust Limits When x = 6, u = 6^2 + 4(6) + 20 = 92. When x = 8, u = 8^2 + 4(8) + 20 = 132. 4. Integrate The integral becomes \int_{92}^{132} \frac{1}{2} \left(\frac{du}{u} - \frac{4dx}{u}\right). The first part is \frac{1}{2} \ln|u| and the second part simplifies to zero since it involves a constant factor over u. 5. Evaluate the Integral Evaluate \frac{1}{2} \ln|u| from 92 to 132: \frac{1}{2} (\ln 132 - \ln 92) = \frac{1}{2} \ln \frac{132}{92}.

Explanation

1. Simplify the Denominator<br /> The denominator $x^2 + 4x + 20$ can be rewritten as $(x+2)^2 + 16$ by completing the square.<br /><br />2. Use Substitution<br /> Let $u = x^2 + 4x + 20$, then $du = (2x + 4)dx$. Rewrite $x dx$ in terms of $du$: $x dx = \frac{1}{2}(du - 4dx)$.<br /><br />3. Adjust Limits<br /> When $x = 6$, $u = 6^2 + 4(6) + 20 = 92$. When $x = 8$, $u = 8^2 + 4(8) + 20 = 132$.<br /><br />4. Integrate<br /> The integral becomes $\int_{92}^{132} \frac{1}{2} \left(\frac{du}{u} - \frac{4dx}{u}\right)$. The first part is $\frac{1}{2} \ln|u|$ and the second part simplifies to zero since it involves a constant factor over $u$.<br /><br />5. Evaluate the Integral<br /> Evaluate $\frac{1}{2} \ln|u|$ from 92 to 132: $\frac{1}{2} (\ln 132 - \ln 92) = \frac{1}{2} \ln \frac{132}{92}$.
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