What is the integral of $int 1/(x^(3/2) + x^(1/2)) dx$ ?

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Answer 1 · 2016-08-19T16:26:46

$2arctan(x^(1/2))+C$

We have:

$I=int1/(x^(3/2)+x^(1/2))dx$

Factor the denominator.

$I=int1/(x^(1/2)(x+1))dx$

Which can be rewritten as:

$I=intx^(-1/2)/(x+1)dx$

$I=intx^(-1/2)/((x^(1/2))^2+1)dx$

Now, let $u=x^(1/2)$ . Thus, $du=1/2x^(-1/2)dx$ .

$I=2int(1/2x^(-1/2))/((x^(1/2))^2+1)dx$

Substituting:

$I=2int1/(u^2+1)du$

This is the arctangent integral:

$I=2arctan(u)+C$

$I=2arctan(x^(1/2))+C$

What is the integral of int 1/(x^(3/2) + x^(1/2)) dxint 1/(x^(3/2) + x^(1/2)) dx?