The value of b > 3 for which \(12∫_3^b \frac{1}{(x^2-1)(x^2-4)}dx=log_e(\frac{40}{40}) \) is equal to.
Correct Answer: 6
Solution and Explanation
The correct answer is: 6
\(l=∫_3^b \frac{1}{(x^2-1)(x^2-4)}dx=\frac{1}{3}(\frac{1}{x^2-4}-\frac{1}{x^2-1})dx\)
\(=ln((\frac{b-2}{b+2})\frac{(b+1)^2}{b-1}^2)-(In\,\frac{4}{5})\)
After simplification ,
\(\frac{49}{40}=\frac{(b-2)}{(b+2)}\frac{(b+1)^2}{(b-1)^2}.\frac{5}{4}\)
⇒ b = 6
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Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
