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Answer :
To solve the integral [tex]\_2^5 9x^6 + \frac{3}{x} dx[/tex], we separate the polynomial and rational functions, find their antiderivatives, and evaluate from 2 to 5, resulting in the final answer of [tex]\frac{9}{7}(5^7 - 2^7) + 3(\ln|5| - \ln|2|).[/tex]
To evaluate the integral [tex]\_2^5 9x^6 + \frac{3}{x} dx < \/p >[/tex]
We start by separating the two terms in the integral and applying basic antiderivative rules:
- For the first term: antiderivative of 9x^6 is [tex]\frac{9}{7}x^7.[/tex]
- For the second term: antiderivative of [tex]\frac{3}{x} is 3\ln|x|.[/tex]
We now evaluate each term from 2 to 5.
- [tex]\frac{9}{7}x^7 \Big|_2^5 = \frac{9}{7}(5^7 - 2^7)[/tex]
- [tex]3\ln|x| \Big|_2^5 = 3(\ln|5| - \ln|2|)[/tex]
Combine these results to write the final answer:
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