অনির্দিষ্ট যোগজীকরণ ( Indefinite integration )

অনুশীলনী \(10.A / Q.3\)-এর প্রশ্নসমুহ
যোজিত ফল নির্ণয় করঃ
\(Q.3.(i)\) \(\int{\frac{a}{x\sqrt{x^2-1}}dx}\)
উত্তরঃ \(a\sec^{-1}{x}+c\)

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

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

\(Q.3.(iv)\) \(\int{\frac{x-1}{\sqrt{x+4}}dx}\)
উত্তরঃ \(\frac{2}{3}(x+4)^{\frac{3}{2}}-10\sqrt{x+4}+c\)

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

\(Q.3.(vi)\) \(\int{\frac{(e^x+1)^2}{\sqrt{e^x}}dx}\)
উত্তরঃ \(\frac{2}{3}e^{\frac{3}{2}x}+4e^{\frac{x}{2}}-2e^{-\frac{x}{2}}+c\)
[ ঢাঃ২০০২ ]

\(Q.3.(vii)\) \(\int{\frac{e^x+1}{\sqrt{e^x}}dx}\)
উত্তরঃ \(2\left(e^{\frac{1}{2}x}-e^{-\frac{1}{2}x}\right)+c\)
[ চঃ২০১৪ ]

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

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

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

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

\(Q.3.(xiii)\) \(\int{\frac{5x^4+4x^2+3}{\sqrt[3]{x}}dx}\)
উত্তরঃ \(\frac{15}{14}x^{\frac{14}{3}}+\frac{3}{2}x^{\frac{8}{3}}+\frac{9}{2}x^{\frac{2}{3}}+c\)

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

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

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

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

\(Q.3.(xviii)\) \(\int{(x^3-5e^x+8)dx}\)
উত্তরঃ \(\frac{1}{4}x^4-5e^x+8x+c\)

\(Q.3.(xix)\) \(\int{(x-2)^3dx}\)
উত্তরঃ \(\frac{1}{4}x^4-2x^3+6x^2-8x+c\)
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