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Evaluate: int1/(x^(1//2)+x^(1//3))\ dx...

Evaluate: `int1/(x^(1//2)+x^(1//3))\ dx`

Text Solution

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`Rightarrow LCM` of `2` and `3=6`
Thus `x=t^{6}` equation `(1)`
`d x=6 t^{5} d t` equation `(2)`
`therefore int frac{d x}{x^{1 / 2}+x^{1 / 3}}=I`
Now `I=int frac{6 t^{2} d t}{t^{3}+t^{2}}`
`=int frac{6 t^{5} d t}{t^{2}(t+1)}`
`=6 int frac{t^{3} d t}{t+1}`
`=6 int frac{(t^{3}+1)-1 d t}{t+1}`
...
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