মূলদ ভগ্নাংশের যোগজীকরণ ( Integration of Rational Fractions )

অনুশীলনী \(10.E / Q.1\)-এর প্রশ্নসমুহ

যোজিত ফল নির্ণয় করঃ
\(Q.1.(i)\) \(\int{\frac{dx}{(x+1)(x+3)}}\)
উত্তরঃ \(\frac{1}{2}\ln{\left|\frac{x+1}{x+3}\right|}+c\)

\(Q.1.(ii)\) \(\int{\frac{dx}{x^2+x}}\)
উত্তরঃ \(\ln{\left|\frac{x}{x+1}\right|}+c\)
[ বঃ২০০৩ ]

\(Q.1.(iii)\) \(\int{\frac{xdx}{(x-a)(x-b)}}\)
উত্তরঃ \(\frac{1}{a-b}\{a\ln{|x-a|}-b\ln{|x-b|}\}+c\)

\(Q.1.(iv)\) \(\int{\frac{x+1}{3x^2-x-2}dx}\)
উত্তরঃ \(\frac{2}{5}\ln{|x-1|}-\frac{1}{15}\ln{|3x+2|}+c\)

\(Q.1.(v)\) \(\int{\frac{x}{x^2-5x+6}dx}\)
উত্তরঃ \(3\ln{|x-3|}-2\ln{|x-2|}+c\)

\(Q.1.(vi)\) \(\int{\frac{1}{x(x-1)(x-3)}dx}\)
উত্তরঃ \(\frac{1}{3}\ln{|x|}+\frac{1}{6}\ln{|x-3|}-\frac{1}{2}\ln{|x-1|}+c\)

\(Q.1.(vii)\) \(\int{\frac{(2x+3)dx}{x^3+x^2-2x}}\)
উত্তরঃ \(\frac{5}{3}\ln{|x-1|}-\frac{3}{2}\ln{|x|}-\frac{1}{6}\ln{|x+2|}+c\)

\(Q.1.(viii)\) \(\int{\frac{2x-1}{x(x-1)(x-2)}dx}\)
উত্তরঃ \(\frac{3}{2}\ln{|x-2|}-\frac{1}{2}\ln{|x|}-\ln{|x-1|}+c\)
[ ঢাঃ২০০৯ ]

\(Q.1.(ix)\) \(\int{\frac{x-3}{(1-2x)(1+x)}dx}\)
উত্তরঃ \(\frac{5}{6}\ln{|1-2x|}-\frac{4}{3}\ln{|x+1|}+c\)
\(Q.1.(x)\) \(\int{\frac{x+1}{x^2-7x+6}dx}\)
উত্তরঃ \(\frac{7}{5}\ln{|x-6|}-\frac{2}{5}\ln{|x-1|}+c\)

\(Q.1.(xi)\) \(\int{\frac{x+1}{x^3+x^2-6x}dx}\)
উত্তরঃ \(\frac{3}{10}\ln{|x-2|}-\frac{1}{6}\ln{|x|}-\frac{2}{15}\ln{|x+3|}+c\)

\(Q.1.(xii)\) \(\int{\frac{x+1}{x^2-7x+10}dx}\)
উত্তরঃ \(2\ln{|x-5|}-\ln{|x-2|}+c\)

\(Q.1.(xiii)\) \(\int{\frac{dx}{x^2+x-6}}\)
উত্তরঃ \(\frac{1}{5}\ln{\left|\frac{x-2}{x+3}\right|}+c\)

\(Q.1.(xiv)\) \(\int{\frac{dx}{(x+1)(x-5)}}\)
উত্তরঃ \(\frac{1}{6}\ln{\left|\frac{x-5}{x+1}\right|}+c\)

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

\(Q.1.(xvi)\) \(\int{\frac{(x+1)dx}{x^2-5x+6}}\)
উত্তরঃ \(4\ln{|x-3|}-3\ln{|x-2|}+c\)

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

\(Q.1.(xviii)\) \(\int{\frac{x}{x^2-5x-6}dx}\)
উত্তরঃ \(\frac{1}{7}\ln{|x+1|}+\frac{6}{7}\ln{|x-6|}+c\)
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