How do you integrate $\int \frac{1}{x^{\frac{3}{2}} + x^{\frac{1}{2}}} d x$ $int 1/(x^(3/2) + x^(1/2)) dx$ ?

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How do you integrate $\int \frac{1}{x^{\frac{3}{2}} + x^{\frac{1}{2}}} d x$ $int 1/(x^(3/2) + x^(1/2)) dx$ ?

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Answer 1 · 2016-09-29T17:15:15

$2 a r c tan \sqrt{x} + C .$ $2arc tansqrtx+C.$

Explanation:

Let $I = \int \frac{1}{x^{\frac{3}{2}} + x^{\frac{1}{2}}} d x = \int \frac{1}{\sqrt{x} (x + 1)} d x$ $I=int1/(x^(3/2)+x^(1/2))dx=int1/{sqrtx(x+1)}dx$ .

We take subst. $\sqrt{x} = t, or, x = t^{2} \Rightarrow d x = 2 t d t .$ $sqrtx=t, or, x=t^2 rArr dx=2tdt.$

$:. I=int1/(t(t^2+1))2tdt=2int1/(t^2+1)dt=2arc tant$

Hence, $I=2arc tansqrtx+C,$ as Respected Eric Sia has derived!

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How do you integrate $\int \frac{1}{x^{\frac{3}{2}} + x^{\frac{1}{2}}} d x$ $int 1/(x^(3/2) + x^(1/2)) dx$ ?

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