The value of b3 for which 12_3^b(dx)/((x^2-1)(x^2-4)) = _e4940 is equal to Answer: 6

Question

The value of $b>3$ for which $12\displaystyle\int_3^b\frac{dx}{(x^2-1)(x^2-4)} = \log_e\dfrac{49}{40}$ is equal to

Answer: 6

Accepted Answer

Partial fractions: $\dfrac1{(x^2-1)(x^2-4)} = \dfrac13\left(\dfrac1{x^2-4}-\dfrac1{x^2-1}\right)$, giving antiderivative

$$\frac1{12}\ln\left|\frac{x-2}{x+2}\right|+\frac1{6}\ln\left|\frac{x-1}{x+1}\right|+C$$

$$12\int_3^b\frac{dx}{(x^2-1)(x^2-4)} = \ln\left|\frac{b-2}{b+2}\cdot\frac{(b+1)^2}{(b-1)^2}\right| - \ln\frac45$$

Setting this equal to $\ln\dfrac{49}{40}$ and solving gives $b=6$.

Solutions · 1