বিশেষ আকারের যোগজীকরণ-১ (Integration of spacial type-1)

অনুশীলনী \(10.C / Q.3\)-এর প্রশ্নসমুহ
যোজিত ফল নির্ণয় করঃ
\(\int{\frac{f^{\prime}(x)}{\sqrt{f(x)}}dx}\) ও \(\int{\frac{f^{\prime}(x)}{f(x)}dx}\) আকার।
\(Q.3.(i)\) \(\int{\frac{x}{\sqrt{1-x^2}}dx}\)
উত্তরঃ \(-\sqrt{1-x^2}+c\)
[ ঢাঃ২০১৪; দিঃ২০১১ ]

\(Q.3.(ii)\) \(\int{\frac{x^3dx}{\sqrt{1-2x^4}}}\)
উত্তরঃ \(-\frac{1}{4}\sqrt{1-2x^4}+c\)
[ চঃ২০১০ ]

\(Q.3.(iii)\) \(\int{\frac{dx}{x\sqrt{1+\ln{|x|}}}}\)
উত্তরঃ \(2\sqrt{1+\ln{|x|}}+c\)
[ কুঃ২০০৩ ]

\(Q.3.(iv)\) \(\int{\frac{\cos{x}}{\sqrt{\sin{x}}}dx}\)
উত্তরঃ \(2\sqrt{\sin{x}}+c\)
[ রাঃ২০১০; কুঃ২০০৫ ]

\(Q.3.(v)\) \(\int{\frac{dx}{\cos^2{x}\sqrt{\tan{x}-1}}}\)
উত্তরঃ \(2\sqrt{\tan{x}-1}+c\)
[ সিঃ২০০২ ]

\(Q.3.(vi)\) \(\int{\frac{\sqrt{\tan{x}}}{\sin{x}\cos{x}}dx}\)
উত্তরঃ \(2\sqrt{\tan{x}}+c\)
[ দিঃ২০১৪; রাঃ,কুঃ২০০৯; সিঃ২০০৭ ]

\(Q.3.(vii)\) \(\int{\frac{dx}{\sqrt{\sin^{-1}{x}}\sqrt{1-x^2}}}\)
উত্তরঃ \(2\sqrt{\sin^{-1}{x}}+c\)

\(Q.3.(viii)\) \(\int{\frac{dx}{(1+x^2)\sqrt{\tan^{-1}{x}+3}}}\)
উত্তরঃ \(2\sqrt{\tan^{-1}{x}+3}+c\)

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

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

\(Q.3.(xi)\) \(\int{\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}dx}\)
উত্তরঃ \(\ln{(e^{x}+e^{-x})}+c\)
[ দিঃ২০১০ ]

\(Q.3.(xii)\) \(\int{\frac{1}{e^{x}+1}dx}\)
উত্তরঃ \(-\ln{(1+e^{-x})}+c\)
[ বঃ২০১৭; যঃ২০১০ ]

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

\(Q.3.(xiv)\) \(\int{\frac{1}{x(1+\ln{|x|})}dx}\)
উত্তরঃ \(\ln{(|1+\ln{|x|}|)}+c\)
[ বঃ২০০৯ ]

\(Q.3.(xv)\) \(\int{\frac{\tan{x}}{\ln{|\cos{x}|}}dx}\)
উত্তরঃ \(-\ln{(\ln{|\cos{x}|)}}+c\)
[ রাঃ২০০৯ ; কুঃ২০০৯,২০০১; সিঃ২০০৯,২০০৭; দিঃ২০০৯; যঃ২০০৮,২০০৪; চঃ,বঃ২০০৬ ]

\(Q.3.(xvi)\) \(\int{\frac{\sin{x}}{1+\cos{x}}dx}\)
উত্তরঃ \(-\ln{|1+\cos{x}|}+c\)

\(Q.3.(xvii)\) \(\int{\frac{\sin{x}}{4-\cos{x}}dx}\)
উত্তরঃ \(\ln{|4-\cos{x}|}+c\)

\(Q.3.(xviii)\) \(\int{\frac{\sin{x}}{3+4\cos{x}}dx}\)
উত্তরঃ \(-\frac{1}{4}\ln{|3+4\cos{x}|}+c\)
[ ঢাঃ২০১৫,২০০৭; বঃ২০১৩ ]

\(Q.3.(xix)\) \(\int{\frac{\sin{x}}{1+2\cos{x}}dx}\)
উত্তরঃ \(-\frac{1}{2}\ln{|1+2\cos{x}|}+c\)
[ রাঃ২০০৩ ]
\(Q.3.(xx)\) \(\int{\frac{1+\cos{x}}{x+\sin{x}}dx}\)
উত্তরঃ \(\ln{|x+\sin{x}|}+c\)
[যঃ২০০৪ ]

\(Q.3.(xxi)\) \(\int{\frac{1-\tan{x}}{1+\tan{x}}dx}\)
উত্তরঃ \(\ln{|\sin{x}+\cos{x}|}+c\)

\(Q.3.(xxii)\) \(\int{\frac{\sec^2{x}}{3-4\tan{x}}dx}\)
উত্তরঃ \(-\frac{1}{4}\ln{|3-4\tan{x}|}+c\)
[ দিঃ২০১১ ]

\(Q.3.(xxiii)\) \(\int{\frac{\sec{x}dx}{\ln{(|\sec{x}+\tan{x}|)}}}\)
উত্তরঃ \(\ln{\{\ln{(|\sec{x}+\tan{x}|)}\}}+c\)

\(Q.3.(xxiv)\) \(\int{\frac{dx}{(1+x^2)\tan^{-1}{x}}}\)
উত্তরঃ \(\ln{|\tan^{-1}{x}|}+c\)
[ সিঃ২০১১; চঃ২০১০; বঃ২০০৪ ]

\(Q.3.(xxv)\) \(\int{\frac{dx}{x(x^4-1)}}\)
উত্তরঃ \(\frac{1}{4}\ln{\left|1-\frac{1}{x^4}\right|}+c\)
[ চুয়েটঃ২০০৭-২০০৮ ]

\(Q.3.(xxvi)\) \(\int{cosec \ {\frac{x}{2}}dx}\)
উত্তরঃ \(2\ln{\left|cosec \ {\frac{x}{2}}-\cot{\frac{x}{2}}\right|}+c\)

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

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

\(Q.3.(xxix)\) \(\int{\frac{\sin{2x}}{a\sin^2{x}+b\cos^2{x}}dx}\)
উত্তরঃ \(\frac{1}{a-b}\ln{|a\sin^2{x}+b\cos^2{x}|}+c\)

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

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

\(Q.3.(xxxii)\) \(\int{\frac{\cos{x}}{\sin{x}+\cos{x}}dx}\)
উত্তরঃ \(\frac{1}{2}x+\frac{1}{2}\ln{|\sin{x}+\cos{x}|}+c\)

\(Q.3.(xxxiii)\) \(\int{\frac{dx}{1+\tan{x}}}\)
উত্তরঃ \(\frac{1}{2}x+\frac{1}{2}\ln{|\sin{x}+\cos{x}|}+c\)
[ ঢাঃ২০১০; কুঃ২০১০; সিঃ২০১৪,২০১০; বঃ২০১৪; যঃ২০১২; দিঃ২০১১ ]

\(Q.3.(xxxiv)\) \(\int{\frac{\cos{x}dx}{(5-2\sin{x})^2}}\)
উত্তরঃ \(\frac{1}{2(5-2\sin{x})}+c\)

\(Q.3.(xxxv)\) \(\int{\frac{\sec^2{(\cot^{-1}{x})}}{1+x^2}dx}\)
উত্তরঃ \(-\frac{1}{x}+c\)
[ বুয়েটঃ২০১০-২০১১ ]

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

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

\(Q.3.(xxxviii)\) \(\int{\frac{dx}{\sqrt{x}+x}}\)
উত্তরঃ \(2\ln{(1+\sqrt{x})}+c\)

\(Q.3.(xxxix)\) \(\int{\frac{dx}{x^{\frac{1}{2}}-x^{\frac{1}{4}}}}\)
উত্তরঃ \(2\sqrt{x}+4\sqrt[4]{x}+4\ln{|\sqrt[4]{x}-1|}+c\)
[ চঃ২০১০,২০১৫; রাঃ২০০৭; যঃ২০০০ ]
1 2 3 4 5

Leave a Reply