7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$A(x - 1) + B(x)(x - 1) + C({x^2}) = x + 1$

Put $x = 0,\,1,\, - 1,$

we get $A = - 1,B = - 2,C = 2$

==> $I = - \int_2^3 {\frac{{dx}}{{{x^2}}} - 2\int_2^3 {\frac{{dx}}{x} + 2\int_2^3 {\frac{{dx}}{{x - 1}}} } } $

==> $I = \left[ {\frac{1}{x}} \right]_2^3 - 2[\log x]_2^3 + 2[\log (x - 1)]_2^3$

==> $I = \frac{1}{3} - \frac{1}{2} - 2\log \frac{3}{2} + 2\log 2$

==> $I = \log \frac{{16}}{9} - \frac{1}{6}$.