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

We have:

Factor the denominator.

Which can be rewritten as:

Substituting:

This is the arctangent integral:

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To find the integral of ( \frac{1}{x^{3/2} + x^{1/2}} ) with respect to ( x ), we can use a substitution. Let's set ( u = x^{1/2} ), then ( du = \frac{1}{2}x^{-1/2}dx ).

Substituting ( u ) for ( x^{1/2} ) and ( du ) for ( \frac{1}{2}x^{-1/2}dx ), we get:

[ \int \frac{1}{x^{3/2} + x^{1/2}} dx = \int \frac{1}{u^3 + u} \cdot 2u , du ]

Now, we have:

[ \int \frac{1}{u^3 + u} \cdot 2u , du = 2\int \frac{1}{u(u^2 + 1)} , du ]

This can be solved using partial fraction decomposition or trigonometric substitution.

Using partial fraction decomposition, we decompose ( \frac{1}{u(u^2 + 1)} ) into partial fractions:

[ \frac{1}{u(u^2 + 1)} = \frac{A}{u} + \frac{Bu + C}{u^2 + 1} ]

Then, solving for ( A ), ( B ), and ( C ) and integrating each term separately yields the final result.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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