int frac{y^2+sqrt[3]{y^4}+sqrt[6]{y^2}}{yleft | vedclass

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(A) ધારો કે $I = \int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}\right)} d y$ $= \int \frac{y^2+y^{4/3}+y^{1/3}}{y(1+y^{2/3})} d y$

Vedclass · Accepted Answer

$\frac{3}{4} \sqrt[3]{y^4}+3 	an ^{-1}(\sqrt[3]{y})+C$

Vedclass · Answer

(A) ધારો કે $I = \int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}ight)} d y$ $= \int \frac{y^2+y^{4/3}+y^{1/3}}{y(1+y^{2/3})} d y$ $= \int \frac{y^{4/3}(y^{2/3}+1)+y^{1/3}}{y(1+y^{2/3})} d y$ $= \int \left(y^{1/3} + \frac{y^{-2/3}}{1+y^{2/3}}ight) d y$ $= \frac{y^{4/3}}{4/3} + \int \frac{y^{-2/3}}{1+(y^{1/3})^2} d y$ $= \frac{3}{4} y^{4/3} + 3 \int \frac{d(y^{1/3})}{1+(y^{1/3})^2}$ $= \frac{3}{4} \sqrt[3]{y^4} + 3 	an^{-1}(y^{1/3}) + C$ $= \frac{3}{4} \sqrt[3]{y^4} + 3 	an^{-1}(\sqrt[3]{y}) + C$

$\int \frac{y^2+\sqrt[3]{y^4}+\sqrt[6]{y^2}}{y\left(1+\sqrt[3]{y^2}\right)} d y=$

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