How do you integrate $int sqrtxsqrt(xsqrtx+1)$ using substitution?

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Answer 1 · 2016-09-27T23:06:50

$(2/3)^2(x^(3/2)-1)^(3/2) + C$

This integral is a power integral. The square root operator helps to conceal de true problem.

$sqrtxsqrt(xsqrtx+1) = x^(1/2)(x^(3/2)-1)^(1/2)=(2/3)^2d/dx(x^(3/2)-1)^(3/2)$

So

$int sqrtx(sqrt(xsqrtx+1))dx = (2/3)^2(x^(3/2)-1)^(3/2) + C$

How do you integrate int sqrtxsqrt(xsqrtx+1)int sqrtxsqrt(xsqrtx+1) using substitution?