Question

Evaluate the definite integral: $∫^3_2 \frac{dx}{x^2-1}$

Answer 1

Let I=$∫^3_2 \frac{dx}{x^2-1}$

$∫\frac{dx}{x^2-1}=\frac{1}{2}log|\frac{x-1}{x+1}|=F(x)$

By second fundamental theorem of calculus,we obtain

$I=F(3)-F(2)$

$=\frac{1}{2}[log|\frac{3-1}{3+1}|-log|\frac{2-1}{2+1|}]$

$=\frac{1}{2}[log|\frac{2}{4}|-log\frac{1}{3}3]$

$=\frac{1}{2}[log\frac{1}{2}-log\frac{1}{3}]$

$=\frac{1}{2}[log\frac{3}{2}]$