মূলদ ভগ্নাংশের যোগজীকরণ ( Integration of Rational Fractions )

অনুশীলনী \(10.E / Q.3\)-এর প্রশ্নসমুহ

যোজিত ফল নির্ণয় করঃ
\(Q.3.(i)\) \(\int{\frac{x^2+3x-4}{x^2-2x-8}dx}\)
উত্তরঃ \(x+4\ln{|x-4|}+\ln{|x+2|}+c\)

\(Q.3.(ii)\) \(\int{\frac{x^2dx}{x^2-4}}\)
উত্তরঃ \(x+\ln{\left|\frac{x-2}{x+2}\right|}+c\)
[ সিঃ,রাঃ,বঃ২০০৮; চঃ২০০২ ]

\(Q.3.(iii)\) \(\int{\frac{x^2-1}{x^2-4}dx}\)
উত্তরঃ \(x+\frac{3}{4}\ln{\left|\frac{x-2}{x+2}\right|}+c\)
[ ঢাঃ২০১১; কুঃ২০০৯,২০০১; যঃ২০০৯; সিঃ২০০৫,২০০৩ ]

\(Q.3.(iv)\) \(\int{\frac{dx}{x^2-9}}\)
উত্তরঃ \(\frac{1}{6}\ln{\left|\frac{x-3}{x+3}\right|}+c\)

\(Q.3.(v)\) \(\int{\frac{x+35}{x^2-25}dx}\)
উত্তরঃ \(4\ln{|x-5|}-3\ln{|x+5|}+c\)
[ চঃ২০০৮; সিঃ২০০৪ ]

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

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

\(Q.3.(viii)\) \(\int{\frac{xdx}{(1-x)^2}}\)
উত্তরঃ \(\ln{|1-x|}+\frac{1}{1-x}+c\)

\(Q.3.(ix)\) \(\int{\frac{6x-10}{(2x+1)^2}dx}\)
উত্তরঃ \(\frac{1}{2}\left\{3\ln{|2x+1|}+\frac{13}{2x+1}\right\}+c\)

\(Q.3.(x)\) \(\int{\frac{x^2+1}{(x+2)^2}dx}\)
উত্তরঃ \(x-4\ln{|x+2|}-\frac{5}{x+2}+c\)
\(Q.3.(xi)\) \(\int{\frac{dx}{e^{2x}-3e^{x}}}\)
উত্তরঃ \(\frac{1}{9}\ln{|e^{x}-3|}-\frac{x}{9}+\frac{1}{3e^{x}}+c\)

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

\(Q.3.(xiii)\) \(\int{\frac{\cos{x}}{(a-\sin{x})(b-\sin{x})}dx}\)
উত্তরঃ \(\frac{1}{a-b}\ln{\left|\frac{a-\sin{x}}{b-\sin{x}}\right|}+c\)

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

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

\(Q.3.(xvi)\) \(\int{\frac{(x-2)^2}{(x+1)^2}dx}\)
উত্তরঃ \(x-6\ln{|x+1|}-\frac{8}{x+1}+c\)

\(Q.3.(xvii)\) \(\int{\frac{\sin{2x}}{3+5\cos{x}}dx}\)
উত্তরঃ \(\frac{2}{25}\left(3\ln{|3+5\cos{x}|}-5\cos{x}\right)+c\)

\(Q.3.(xviii)\) \(\int{\frac{dx}{(e^x-1)(e^x+3)}}\)
উত্তরঃ \(\frac{1}{12}\ln{|e^x+3|}+\frac{1}{4}\ln{|e^x-1|}-\frac{x}{3}+c\)

\(Q.3.(xix)\) \(\int{\frac{x^2dx}{x^2-16}}\)
উত্তরঃ \(x+2\ln{\left|\frac{x-4}{x+4}\right|}+c\)
1 2 3 4 5

Leave a Reply