intigration of Rational Algebraic Fractions (মূলদ ভগ্নাংশ)
∫dx(x−1)(x−2)=?\int\frac{dx}{\left(x-1\right)\left(x-2\right)}=? ∫(x−1)(x−2)dx=?
ln∣x−3∣+c\ln{\left|x-3\right|+c} ln∣x−3∣+c
ln∣x−2∣+ln∣x−1∣+c\ln{\left|x-2\right|+\ln{\left|x-1\right|+c}} ln∣x−2∣+ln∣x−1∣+c
ln∣x−4∣+c\ln{\left|x-4\right|+c} ln∣x−4∣+c
ln∣x−2x−1∣+c\ln{\left|\frac{x-2}{x-1}\right|}+c lnx−1x−2+c
1(x−1)(x−2)≡1x−2−1x−1∴ ∫1(x−1)(x−2)dx=ln∣x−2∣−ln∣x−1∣+c\frac{1}{\left(x-1\right)\left(x-2\right)}\equiv\frac{1}{x-2}-\frac{1}{x-1} \\\\\therefore\ \int{\frac{1}{\left(x-1\right)\left(x-2\right)}dx=\ln{\left|x-2\right|-\ln{\left|x-1\right|+c}}} (x−1)(x−2)1≡x−21−x−11∴ ∫(x−1)(x−2)1dx=ln∣x−2∣−ln∣x−1∣+c
