মূলদ ভগ্নাংশের যোগজীকরণ ( Integration of Rational Fractions )

অনুশীলনী \(10.E\) উদাহরণ সমুহ
যোজিত ফল নির্ণয় করঃ
\((1.)\) \(\int{\frac{x}{x^2-5x+6}dx}\)
উত্তরঃ \(3\ln{|x-3|}-2\ln{|x-2|}+c\)

\((2.)\) \(\int{\frac{x-1}{(x-2)(x-3)}dx}\)
উত্তরঃ \(2\ln{|x-3|}-\ln{|x-2|}+c\)

\((3.)\) \(\int{\frac{2x+1}{(x+2)(x-3)^2}dx}\)
উত্তরঃ \(\frac{3}{25}\ln{|x-3|}-\frac{3}{25}\ln{|x+2|}-\frac{7}{5(x-3)}+c\)

\((4.)\) \(\int{\frac{x}{(x-1)(x^2+4)}dx}\)
উত্তরঃ \(\frac{1}{5}\ln{|x-1|}-\frac{1}{10}\ln{|x^2+4|}+\frac{2}{5}\tan^{-1}{\left(\frac{x}{2}\right)}+c\)

\((5.)\) \(\int{\frac{x^2+3x-4}{x^2-2x-8}dx}\)
উত্তরঃ \(x+4\ln{|x-4|}+\ln{|x+2|}+c\)

\((6.)\) \(\int{\frac{x^2}{x^2-4}dx}\)
উত্তরঃ \(x+\ln{\left|\frac{x-2}{x+2}\right|}+c\)
\((7.)\) \(\int{\frac{x^2+x+1}{x^2-x+1}dx}\)
উত্তরঃ \(x+\ln{|x^2-x+1|}+\frac{2}{\sqrt{3}}\tan^{-1}{\left(\frac{2x-1}{\sqrt{3}}\right)}+c\)

\((8.)\) \(\int{\frac{x+1}{(x-2)(x+3)}dx}\)
উত্তরঃ \(\frac{1}{5}\left(3\ln{|x-2|}+2\ln{|x+3|}\right)+c\)

\((9.)\) \(\int{\frac{1}{x^2+5x+4}dx}\)
উত্তরঃ \(\frac{1}{3}\ln{\left|\frac{x+1}{x+4}\right|}+c\)

\((10.)\) \(\int{\frac{x^3-2x+3}{x^2+x-2}dx}\)
উত্তরঃ \(\frac{x}{2}-x+\frac{1}{3}\ln{|x+2|}+\frac{2}{3}\ln{|x-1|}+c\)

\((11.)\) \(\int{\frac{x^2dx}{(x^2+4)(x^2-3)}}\)
উত্তরঃ \(\frac{2}{7}\tan^{-1}{\left(\frac{x}{2}\right)}+\frac{\sqrt{3}}{14}\ln{\left|\frac{x-\sqrt{3}}{x+\sqrt{3}}\right|}+c\)
1 2 3 4 5

Leave a Reply