We have to solve the following integration . $\int \frac{\sqrt x+1}{\sqrt x(\sqrt[3]x+1)}\text{d}x$
I am not getting any idea how to solve it .
I am not getting anything to substitute
Evaluating integration
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integration
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0try getting rid of the radicals by noticing that the lowest multiple of powers to achieve this is $2\times 3$ or $x = t^6$. try this sub then come back. – 2017-01-04
1 Answers
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Hint:
We have $$I =\frac {1}{6x^{\frac {5}{6}}} \int \frac {6(\sqrt{x}+1)(\sqrt [3]{x})}{\sqrt [3]{x}+1} \mathrm{d}x $$ Now if $u=\sqrt [6]{x}$ then we have $$I =\int \frac {u^2 (u^3+1)}{u^2+1} du =\int (\frac {u-1}{u^2+1} + u^3-u+1) \mathrm{d}u $$ Hope you can take it from here.