Question

integrate x/(x ^ 2 - 1) dx from 2 to 3​

Answer 1

$\displaystyle\boxed{\int_{2}^{3} \dfrac{x}{x^{2}-1}dx=\dfrac{1}{2} log\dfrac{8}{3}}$

Step-by-step explanation:

Now, $\displaystyle\int_{2}^{3} \dfrac{x}{x^{2}-1}dx$

$\displaystyle=\dfrac{1}{2} \int_{2}^{3} \dfrac{2x}{x^{2}-1}dx$

$\displaystyle=\dfrac{1}{2} \int_{2}^{3} \dfrac{d(x^{2}-1)}{x^{2}-1}dx$

$\displaystyle=\dfrac{1}{2} [log(x^{2}-1)]_{2}^{3}$

$\displaystyle=\dfrac{1}{2} [log(3^{2}-1)-log(2^{2}-1)]$

$\displaystyle=\dfrac{1}{2} [log(9-1)-log(4-1)]$

$\displaystyle=\dfrac{1}{2} [log8-log3]$

$\displaystyle=\dfrac{1}{2} log\dfrac{8}{3}$

This is the required integral.

Remember:

$\displaystyle\int \dfrac{d(f(x))}{f(x)}=log((f(x))$