বিশেষ আকারের যোগজীকরণ-২ (Integration of spacial type-2)

অনুশীলনী \(10.D / Q.2\)-এর প্রশ্নসমুহ
যোজিত ফল নির্ণয় করঃ
\(Q.2.(i)\) \(\int{\frac{xdx}{x^4+1}}\)
উত্তরঃ \(\frac{1}{2}\tan^{-1}{(x^2)}+c\)
[ বঃ২০১১; রাঃ২০০৮ ]

\(Q.2.(ii)\) \(\int{\frac{xdx}{x^4-4}}\)
উত্তরঃ \(\frac{1}{8}\ln{\left|\frac{x^2-2}{x^2+2}\right|}+c\)

\(Q.2.(iii)\) \(\int{\frac{xdx}{x^4+3}}\)
উত্তরঃ \(\frac{1}{2\sqrt{3}}\tan^{-1}{\left(\frac{x^2}{\sqrt{3}}\right)}+c\)

\(Q.2.(iv)\) \(\int{\frac{3x^2}{1+x^6}dx}\)
উত্তরঃ \(\tan^{-1}{(x^3)}+c\)
[ চঃ২০০৮ ]

\(Q.2.(v)\) \(\int{\frac{5e^{2x}}{1+e^{4x}}dx}\)
উত্তরঃ \(\frac{5}{2}\tan^{-1}{\left(e^{2x}\right)}+c\)
[ চঃ২০১১, ২০০১ ]

\(Q.2.(vi)\) \(\int{\frac{1}{e^x+e^{-x}}dx}\)
উত্তরঃ \(\tan^{-1}{\left(e^{x}\right)}+c\)
[ সিঃ২০১০; বঃ২০০৯,২০০৭,২০০৫; চঃ২০০৮; রাঃ২০০৭; ঢাঃ২০০৬; যঃ২০০৩ ]

\(Q.2.(vii)\) \(\int{\frac{x^2}{e^{x^3}+e^{-x^3}}dx}\)
উত্তরঃ \(\frac{1}{3}\tan^{-1}{\left(e^{x^3}\right)}+c\)
[ কুয়েটঃ২০১০-২০১১ ]

\(Q.2.(viii)\) \(\int{\frac{\cos{2x}dx}{\sin^2{2x}+8}}\)
উত্তরঃ \(\frac{\sqrt{2}}{8}\tan^{-1}{\left(\frac{\sin{2x}}{2\sqrt{2}}\right)}+c\)

\(Q.2.(ix)\) \(\int{\frac{d\theta}{1+3\cos^2{\theta}}}\)
উত্তরঃ \(\frac{1}{2}\tan^{-1}{\left(\frac{1}{2}\tan{\theta}\right)}+c\)
[ চঃ২০১৩,২০০৯; রাঃ২০০৭; বঃ২০০৫; ঢাঃ২০১২; কুঃ২০১৫; বুটেক্সঃ২০০২-২০০৩ ]

\(Q.2.(x)\) \(\int{\frac{dx}{1+\cos^2{x}}}\)
উত্তরঃ \(\frac{1}{\sqrt{2}}\tan^{-1}{\left(\frac{\tan{x}}{\sqrt{2}}\right)}+c\)
[ রাঃ২০০৬ ]

\(Q.2.(xi)\) \(\int{\frac{x^2+1}{x^4+1}dx}\)
উত্তরঃ \(\frac{1}{\sqrt{2}}\tan^{-1}{\left(\frac{x^2-1}{\sqrt{2}x}\right)}+c\)
[ বুয়েটঃ২০১৪-২০১৫ ]

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

\(Q.2.(xiii)\) \(\int{\frac{x^2dx}{\sqrt{1-x^6}}}\)
উত্তরঃ \(\frac{1}{3}\sin^{-1}{(x^3)}+c\)
[ যঃ২০১১; বঃ২০০৮; ঢাঃ২০০২]

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

\(Q.2.(xv)\) \(\int{\frac{1}{e^x-e^{-x}}dx}\)
উত্তরঃ \(\frac{1}{2}\ln{\left|\frac{e^x-1}{e^x+1}\right|}+c\)
[ বুয়েটঃ২০০৫ ]

\(Q.2.(xvi)\) \(\int{\frac{x^2}{e^{x^3}-e^{-x^3}}dx}\)
উত্তরঃ \(\frac{1}{6}\ln{\left|\frac{e^{x^3}-1}{e^{x^3}+1}\right|}+c\)
[ বুয়েটঃ২০০১-২০০২ ]

\(Q.2.(xvii)\) \(\int{\frac{\cos{x}dx}{3+\cos^2{x}}}\)
উত্তরঃ \(\frac{1}{4}\ln{\left|\frac{2+\sin{x}}{2-\sin{x}}\right|}+c\)
[ কুয়েটঃ২০০৫-২০০৬; বুটেক্সঃ২০০৬-২০০৭ ]

\(Q.2.(xviii)\) \(\int{\frac{e^x}{\sqrt{e^{2x}+1}}dx}\)
উত্তরঃ \(\ln{\left|e^x+\sqrt{e^{2x}+1}\right|}+c\)
\(Q.2.(xix)\) \(\int{\frac{dx}{(a^2+x^2)^{\frac{3}{2}}}}\)
উত্তরঃ \(\frac{x}{a^2\sqrt{a^2+x^2}}+c\)
[ রুয়েটঃ২০০৬-২০০৭; যঃ২০০২ ]

\(Q.2.(xx)\) \(\int{\frac{dx}{(x^2+9)^{2}}}\)
উত্তরঃ \(\frac{1}{108}\left[2\tan^{-1}{\left(\frac{x}{3}\right)}+\sin{\left\{2\tan^{-1}{\left(\frac{x}{3}\right)}\right\}}\right]+c\)
[ বুয়েটঃ২০০০-২০০১ ]

\(Q.2.(xxi)\) \(\int{\frac{dx}{x\sqrt{x^2+1}}}\)
উত্তরঃ \(\ln{\left|\frac{\sqrt{1+x^2}-1}{x}\right|}+c\)
[ চুয়েটঃ২০১০-২০১১ ]

\(Q.2.(xxii)\) \(\int{\frac{dx}{a^2\sin^2{x}+b^2\cos^2{x}}}\)
উত্তরঃ \(\frac{1}{ab}\tan^{-1}{\left(\frac{a\tan{x}}{b}\right)}+c\)

\(Q.2.(xxiii)\) \(\int{\frac{dx}{a^2\cos^2{x}-b^2\sin^2{x}}}\)
উত্তরঃ \(\frac{1}{2ab}\ln{\left|\frac{a+b\tan{x}}{a-b\tan{x}}\right|}+c\)

\(Q.2.(xxiv)\) \(\int{\frac{dx}{25\sin^2{x}-16\cos^2{x}}}\)
উত্তরঃ \(\frac{1}{40}\ln{\left|\frac{5\tan{x}-4}{5\tan{x}+4}\right|}+c\)

\(Q.2.(xxv)\) \(\int{\frac{dx}{a^2+b^2\sin^2{x}}}\)
উত্তরঃ \(\frac{1}{a\sqrt{a^2+b^2}}\tan^{-1}{\left(\frac{\sqrt{a^2+b^2}}{a}\tan{x}\right)}+c\)

\(Q.2.(xxvi)\) \(\int{\frac{e^x}{1+e^{2x}}dx}\)
উত্তরঃ \(\tan^{-1}{\left(e^x\right)}+c\)

\(Q.2.(xxvii)\) \(\int{\frac{\sin{8x}}{9+\sin^4{4x}}dx}\)
উত্তরঃ \(\frac{1}{12}\tan^{-1}{\left(\frac{1}{3}\sin^2{4x}\right)}+c\)

\(Q.2.(xxviii)\) \(\int{\frac{\sec^2{x}}{4+9\tan^2{x}}dx}\)
উত্তরঃ \(\frac{1}{6}\tan^{-1}{\left(\frac{3}{2}\tan{x}\right)}+c\)

\(Q.2.(xxix)\) \(\int{\frac{\cos{x}}{9+\sin^2{x}}dx}\)
উত্তরঃ \(\frac{1}{3}\tan^{-1}{\left(\frac{1}{3}\sin{x}\right)}+c\)

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

\(Q.2.(xxxi)\) \(\int{\frac{x+35}{x^2-25}dx}\)
উত্তরঃ \(\frac{1}{2}\ln{\left|x^2-25\right|}+\frac{7}{2}\ln{\left|\frac{x-5}{x+5}\right|}+c\)

\(Q.2.(xxxii)\) \(\int{\frac{x^2}{x^2-4}dx}\)
উত্তরঃ \(x+\ln{\left|\frac{x-2}{x+2}\right|}+c\)
[ রাঃ২০১১; বঃ২০০৮ ]

\(Q.2.(xxxiii)\) \(\int{\frac{dx}{x\sqrt{x^4-1}}}\)
উত্তরঃ \(\frac{1}{2}\sec^{-1}{(x^2)}+c\)

\(Q.2.(xxxiv)\) \(\int{\frac{x^2-1}{x^2-4}dx}\)
উত্তরঃ \(x+\frac{3}{4}\ln{\left|\frac{x-2}{x+2}\right|}+c\)
[ ঢাঃ২০১৫,২০১১; সিঃ২০১২; বঃ২০১৩ ]

\(Q.2.(xxxv)\) \(\int{\frac{dx}{(a^2-x^2)^{\frac{3}{2}}}}\)
উত্তরঃ \(\frac{x}{a^2\sqrt{a^2-x^2}}+c\)

\(Q.2.(xxxvi)\) \(\int{\frac{x^2dx}{x^{4}+a^{4}}}\)
উত্তরঃ \(\frac{1}{2\sqrt{2}a}\left\{\tan^{-1}{\left(\frac{x^2-a^2}{\sqrt{2}ax}\right)}
+\frac{1}{2}\ln{\left|\frac{x^2+a^2-\sqrt{2}ax}{x^2+a^2+\sqrt{2}ax}\right|}\right\}+c\)

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