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Question
Integrate the function:
`1/(x^(1/2) + x^(1/3)) ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))), "put x" = t^6]`
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Solution
Let I = `int 1/(x^(1/2) + x^(1/3))`dx
Put `x^(1/6) = t, x = t^6 `
dx = 6 t5 dt
`int = (6 t^5 dt)/(t^3 + t^2) = 6 int (t^3/(t + 1))`dt
`= 6 int (t^3 + 1 - 1)/(t + 1)`dt
`= 6 int ((t^3 + 1)/(t + 1) - 1/(t + 1)) dt`
`= int (((t + 1)(t^2 - t - 1))/(t + 1) - 1/(t + 1)) dt`
`= 6 int (t^2 - t + 1 - 1/(t + 1)) dt`
`= 6 int t^2 dt - 6 int t dt - 6 int 1/(t + 1) dt`
`= (6t^3)/3 - (6t^2)/2 - 6 log |t + 1| + C`
= 2t3 - 3t2 - 6 log |t + 1| + C
= `2sqrtx - 3x^(1/3) + 6x^(1/6) - 6 log |x^(1/6) + 1| + C`
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