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

অনুশীলনী \(10.D / Q.1\)-এর প্রশ্নসমুহ

যোজিত ফল নির্ণয় করঃ
\(Q.1.(i)\) \(\int{\frac{dx}{16+x^2}}\)
উত্তরঃ \(\frac{1}{4}\tan^{-1}{\left(\frac{x}{4}\right)}+c\)

\(Q.1.(ii)\) \(\int{\frac{1}{16a^2+x^2}dx}\)
উত্তরঃ \(\frac{1}{4a}\tan^{-1}{\left(\frac{x}{4a}\right)}+c\)

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

\(Q.1.(iv)\) \(\int{\frac{1}{9x^2+4}dx}\)
উত্তরঃ \(\frac{1}{6}\tan^{-1}{\left(\frac{3x}{2}\right)}+c\)
[ বঃ২০০৭; সিঃ২০০২; কুঃ২০০০ ]

\(Q.1.(v)\) \(\int{\frac{dx}{\sqrt{25-x^2}}}\)
উত্তরঃ \(\sin^{-1}{\left(\frac{x}{5}\right)}+c\)
[ দিঃ২০১০; চঃ ২০০৩ ]

\(Q.1.(vi)\) \(\int{\frac{dx}{\sqrt{25-16x^2}}}\)
উত্তরঃ \(\frac{1}{4}\sin^{-1}{\left(\frac{4x}{5}\right)}+c\)
[সিঃ ২০০৯ ]

\(Q.1.(vii)\) \(\int{\frac{dx}{\sqrt{5-4x^2}}}\)
উত্তরঃ \(\frac{1}{2}\sin^{-1}{\left(\frac{2x}{\sqrt{5}}\right)}+c\)
[ যঃ ২০১১; চঃ২০১১,২০০৩; বঃ ২০১০,২০০৬; ঢাঃ২০০৯; রাঃ২০০৮ ]

\(Q.1.(viii)\) \(\int{\frac{dx}{\sqrt{2-3x^2}}}\)
উত্তরঃ \(\frac{1}{\sqrt{3}}\sin^{-1}{\left(\sqrt{\frac{3}{2}}x\right)}+c\)
[ যঃ ২০০৫; কুঃ২০১০,২০০৭ ]

\(Q.1.(ix)\) \(\int{\frac{dx}{\sqrt{9-16x^2}}}\)
উত্তরঃ \(\frac{1}{4}\sin^{-1}{\left(\frac{4x}{3}\right)}+c\)
[ যঃ ২০০৫; কুঃ২০১০,২০০৭; ঢাঃ২০০৪,২০০৬; রাঃ২০০৩,২০০৬ ]

\(Q.1.(x)\) \(\int{\frac{dx}{\sqrt{12-16x^2}}}\)
উত্তরঃ \(\frac{1}{4}\sin^{-1}{\left(\frac{2x}{\sqrt{3}}\right)}+c\)
[ বঃ২০১৭ ]

\(Q.1.(xi)\) \(\int{\frac{dx}{16-4x^2}}\)
উত্তরঃ \(\frac{1}{16}\ln{\left|\frac{2+x}{2-x}\right|}+c\)
[ সিঃ২০০১ ]

\(Q.1.(xii)\) \(\int{\frac{dx}{9-4x^2}}\)
উত্তরঃ \(\frac{1}{12}\ln{\left|\frac{3+2x}{3-2x}\right|}+c\)
[ সিঃ২০০১ ]

\(Q.1.(xiii)\) \(\int{\sqrt{1-9x^2}dx}\)
উত্তরঃ \(\frac{x\sqrt{1-9x^2}}{2}+\frac{1}{6}\sin^{-1}{(3x)}+c\)
[ বঃ২০০১ ]
\(Q.1.(xiv)\) \(\int{\frac{1}{2x^2+x+1}dx}\)
উত্তরঃ \(\frac{2}{\sqrt{7}}\tan^{-1}{\left(\frac{4x+1}{\sqrt{7}}\right)}+c\)

\(Q.1.(xv)\) \(\int{\frac{1}{x^2-x+1}dx}\)
উত্তরঃ \(\frac{2}{\sqrt{3}}\tan^{-1}{\left(\frac{2x-1}{\sqrt{3}}\right)}+c\)
[ চঃ২০০৩ ]

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

\(Q.1.(xvii)\) \(\int{\frac{dx}{\sqrt{2ax-x^2}}}\)
উত্তরঃ \(\sin^{-1}{\left(\frac{x-a}{a}\right)}+c\)
[ যঃ২০০৯]

\(Q.1.(xviii)\) \(\int{\frac{dx}{\sqrt{15-4x-4x^2}}}\)
উত্তরঃ \(\frac{1}{2}\sin^{-1}{\left(\frac{2x+1}{4}\right)}+c\)

\(Q.1.(xix)\) \(\int{\frac{1}{5+4x-x^2}dx}\)
উত্তরঃ \(\frac{1}{6}\ln{\left|\frac{x+1}{5-x}\right|}+c\)
[ কুয়েটঃ২০০৪-২০০৫]

\(Q.1.(xx)\) \(\int{\frac{1}{\sqrt{x^2+4x+13}}dx}\)
উত্তরঃ \(\ln{\left|(x+2)+\sqrt{x^2+4x+13}\right|}+c\)
[ রাঃ২০০২]

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

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

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

\(Q.1.(xxiv)\) \(\int{\frac{1}{x^2+4x+13}dx}\)
উত্তরঃ \(\frac{1}{3}\tan^{-1}{\left(\frac{x+2}{3}\right)}+c\)

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

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

\(Q.1.(xxvii)\) \(\int{\sqrt{16-9x^2}dx}\)
উত্তরঃ \(\frac{x\sqrt{16-9x^2}}{2}+\frac{8}{3}\sin^{-1}{\left(\frac{3x}{4}\right)}+c\)

\(Q.1.(xxviii)\) \(\int{\sqrt{a^2-x^2}dx}\)
উত্তরঃ \(\frac{a^2}{2}\sin^{-1}{\left(\frac{x}{a}\right)}+\frac{x\sqrt{a^2-x^2}}{2}+c\)
1 2 3 4 5