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I want to evaluate the following integral $$\int \frac{1}{4x^{1/2} + x^{3/2} } dx$$ I tried using substitution but that didn't help. I just want a hint. Thank you in advance.

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closed as off-topic by RRL, Shogun, Cesareo, Jendrik Stelzner, user10354138 Jun 8 at 16:41

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Let $$x=t^2$$

$$dx=2tdt$$

$$\int \frac{1}{4x^{1/2} + x^{3/2} } dx= \int \frac{2tdt}{4t+t^3} =\int \frac{2dt}{4+t^2} $$

That is a well known integral.

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Hint:

Rewrite your integral as:

$$ \int \frac{1}{\sqrt{x}( 4 + x)} dx $$

Now can you solve it using substitution! What substitution do you think is the best in this case?

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    $\begingroup$ Yes! If I pick $u = \sqrt{x}$ then I will get a very nice form of $\int \frac{1}{4+u^2} $ which can be solve using $\arctan$. Thank you. $\endgroup$ – MathStudent Jun 7 at 17:18
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    $\begingroup$ Don't forget the 2 in-front of the $\int \frac{1}{4+u^2}$ since the derivative of $\sqrt{x} = \frac{1}{2 \sqrt{x} }$ $\endgroup$ – user209663 Jun 7 at 17:19
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Hint: Substitute $$x=t^2$$ in your integral

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    $\begingroup$ I see now!!! Thank you $\endgroup$ – MathStudent Jun 7 at 17:16

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