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

অনুশীলনী \(10.D\) উদাহরণ সমুহ
যোজিত ফল নির্ণয় করঃ
\((1.)\) \(\int{\frac{dx}{x^2+25}}\)
উত্তরঃ \(\frac{1}{5}\tan^{-1}{\left(\frac{x}{5}\right)}+c\)

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

\((3.)\) \(\int{\frac{1}{9x^2-16}dx}\)
উত্তরঃ \(\frac{1}{24}\ln{|\frac{3x-4}{3x+4}|}+c\)

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

\((5.)\) \(\int{\frac{xdx}{2x^4-3x^2-2}}\)
উত্তরঃ \(\frac{1}{10}\ln{\left|\frac{x^2-2}{2x^2+1}\right|}+c\)

\((6.)\) \(\int{\frac{e^xdx}{5-4e^x-e^{2x}}}\)
উত্তরঃ \(\frac{1}{6}\ln{\left|\frac{5+e^x}{1-e^x}\right|}+c\)

\((7.)\) \(\int{\frac{(3x-5)dx}{x^2-2x+10}}\)
উত্তরঃ \(\frac{3}{2}\ln{\left|x^2-2x+10\right|}-\frac{2}{3}\tan^{-1}{\left(\frac{x-1}{3}\right)}+c\)

\((8.)\) \(\int{\frac{dx}{\sqrt{2x^2+3x+4}}}\)
উত্তরঃ \(\frac{1}{\sqrt{2}}\ln{\left|x+\frac{3}{4}+\sqrt{x^2+\frac{3}{2}x+2}\right|}+c\)

\((9.)\) \(\int{\frac{dx}{\sqrt{3-5x-2x^2}}}\)
উত্তরঃ \(\frac{1}{\sqrt{2}}\sin^{-1}{\left(\frac{4x+5}{7}\right)}+c\)
\((10.)\) \(\int{\frac{(x+1)}{\sqrt{4+8x-5x^2}}dx}\)
উত্তরঃ \(\frac{9}{5\sqrt{5}}\sin^{-1}{\left(\frac{5x-4}{6}\right)}-\frac{1}{5}\sqrt{4+8x-5x^2}+c\)

\((11.)\) \(\int{\frac{dx}{(2x-3)\sqrt{3x+2}}}\)
উত্তরঃ \(-\frac{1}{\sqrt{26}}\ln{\left|\frac{\sqrt{6x+4}-\sqrt{13}}{\sqrt{6x+4}+\sqrt{13}}\right|}+c\)

\((12.)\) \(\int{\frac{dx}{(x-3)\sqrt{2x^2-12x+17}}}\)
উত্তরঃ \(-\sin^{-1}{\left(\frac{1}{\sqrt{2}(x-3)}\right)}+c\)

\((13.)\) \(\int{\frac{dx}{(2x^2+a^2)\sqrt{x^2+a^2}}}\)
উত্তরঃ \(\frac{1}{a^2}\tan^{-1}{\left(\frac{x}{\sqrt{x^2+a^2}}\right)}+c\)

\((14.)\) \(\int{\frac{1}{(x-b)^3(x-a)^2}dx}\)
উত্তরঃ \(\frac{1}{(a-b)^4}\left[\frac{3(x-a)}{x-b}+3\ln{\left|\frac{x-b}{x-a}\right|}-\frac{(x-a)^2}{2(x-b)^2}-\frac{x-b}{x-a}\right]+c\)

\((15.)\) \(\int{\frac{dx}{9x^2+4}}\)
উত্তরঃ \(\frac{1}{6}\tan^{-1}{\left(\frac{3x}{2}\right)}+c\)

\((16.)\) \(\int{\frac{dx}{\sqrt{5-4x^2}}}\)
উত্তরঃ \(\sin^{-1}{\left(\frac{\sqrt{2}x}{\sqrt{5}}\right)}+c\)

\((17.)\) \(\int{\frac{dx}{16x^2-9}}\)
উত্তরঃ \(\frac{1}{24}\ln{\left|\frac{4x-3}{4x+2}\right|}+c\)
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