Find the Integration Value: 1 ∫(((√x)d(x))/(1+^3 (√x)))=? 2 ∫(dx/(x^(1/2) −x^(1/4) ))=?


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Question Number 140330 by Satyendra last updated on 06/May/21

$F i n d t h e I n t e g r a t i o n V a l u e :$ $1 \int \frac{\sqrt{x} d (x)}{1 +^{3} \sqrt{x}} = ?$ $2 \int \frac{d x}{x^{\frac{1}{2}} - x^{\frac{1}{4}}} = ?$
	Answered by john_santu last updated on 06/May/21

	$(1) l e t x = t^{6} \to d x = 6 t^{5} d t$ $I = \int \frac{t^{3}}{1 + t^{2}} (6 t^{5} d t)$ $I = \int \frac{6 t^{8}}{1 + t^{2}} d t .$ $n o w i t e a s y t o s o l v e$
	Answered by benjo_mathlover last updated on 06/May/21
	$(2) I= ∫ (dx/( (√x)−(x)^(1/4) )) let x = z^4 →dx = 4z^3 dz I= ∫ ((4z^3 )/(z^2 −z)) dz = ∫ ((4z^2 )/(z−1)) dz use partial fraction$
	$(2) I = \int \frac{dx}{\sqrt{x} - \sqrt[4]{x}}$ $let x = z^{4} \to dx = {4 z}^{3} dz$ $I = \int \frac{{4 z}^{3}}{z^{2} - z} dz = \int \frac{{4 z}^{2}}{z - 1} dz$ $use partial fraction$
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