x x+a - x+b dx = x x+a - x+b x+a + x+b x+a + x+b dx = x( x+a + x+b ) ( x+a )^2-( x+b )^2 dx = x( x+a + x+b ) x+a-x-b dx =1 a-b ( x+a + x+b )dx =1 a-b [ ( x+a )dx+ ( x+b )dx ] =1 a-b [ (x+a-a)( x+a )dx+ (x+b-b)( x+b )dx =1 a-b [ (x+a)( x+a )dx-a ( x+a )dx + (x+b)( x+b )dx-b ( x+b )dx ] =1 a-b [ (x+a)^3 2 dx-a (x+a)^1 2 dx+ (x+b)^3 2 dx-b (x+b)^1 2 dx ] =1 a-b [ (x+a)^5 2 5 2 -a (x+a)^3 2 3 2 + (x+b)^5 2 5 2 -b (x+b)^3 2 3 2 +c where, c is an arbitrary constant. =1 a-b [2 5 (x+a)^5 2 - 2a 3 (x+a)^3 2 +2 5 (x+b)^5 2 - 2b 3 (x+b)^3 2 ]+c where, c is an arbitrary constant. Hence, x x+a - x+b dx= 1 a-b [2 5 (x+a)^5 2 - 2a 3 (x+a)^3 2 +2 5 (x+b)^5 2 - 2b 3 (x+b)^3 2 ]+c where, c is an arbitrary constant.

x x+a - x+b dx - Vidyadip

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Question

$\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}$

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Answer

$\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}$
$=\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\times\frac{\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}}}{\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}}}\text{dx}$
$=\int\frac{\text{x}(\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}})}{(\sqrt{\text{x}+\text{a}})^2-(\sqrt{\text{x}+\text{b}})^2}\text{dx}$
$=\int\frac{\text{x}(\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}})}{\text{x}+\text{a}-\text{x}-\text{b}}\text{dx}$
$=\frac{1}{\text{a}-\text{b}}\int\text{x}(\sqrt{\text{x}+\text{a}}+\sqrt{\text{x}+\text{b}})\text{dx}$
$=\frac{1}{\text{a}-\text{b}}\big[\int\text{x}(\sqrt{\text{x}+\text{a}})\text{dx}+\int\text{x}(\sqrt{\text{x}+\text{b}})\text{dx}\big]$
$=\frac{1}{\text{a}-\text{b}}\big[\int(\text{x}+\text{a}-\text{a})(\sqrt{\text{x}+\text{a}})\text{dx}+\int(\text{x}+\text{b}-\text{b})(\sqrt{x+\text{b}})\text{dx}$
$=\frac{1}{\text{a}-\text{b}}\big[\int(\text{x}+\text{a})(\sqrt{\text{x}+\text{a}})\text{dx}-\text{a}\int(\sqrt{\text{x}+\text{a}})\text{dx}\\+\int(\text{x}+\text{b})(\sqrt{\text{x}+\text{b}})\text{dx}-\text{b}\int(\sqrt{\text{x}+\text{b}})\text{dx}\big]$
$=\frac{1}{\text{a}-\text{b}}\big[\int(\text{x}+\text{a})^\frac{3}{2}\text{dx}-\text{a}\int(\text{x}+\text{a})^\frac{1}{2}\text{dx}+\int(\text{x}+\text{b})^\frac{3}{2}\text{dx}-\text{b}\int(\text{x}+\text{b})^\frac{1}{2}\text{dx}\big]$
$=\frac{1}{\text{a}-\text{b}}\Big[\frac{(\text{x}+\text{a})^\frac{5}{2}}{\frac{5}{2}}-\text{a}\frac{(\text{x}+\text{a})^\frac{3}{2}}{\frac{3}{2}}+\frac{(\text{x}+\text{b})^\frac{5}{2}}{\frac{5}{2}}-\text{b}\frac{(\text{x}+\text{b})^\frac{3}{2}}{\frac{3}{2}}+\text{c}$ where, c is an arbitrary constant.
$=\frac{1}{\text{a}-\text{b}}\Big[\frac{2}{5}(\text{x}+\text{a})^\frac{5}{2}-\frac{2\text{a}}{3}(\text{x}+\text{a})^\frac{3}{2}+\frac{2}{5}(\text{x}+\text{b})^\frac{5}{2}-\frac{2\text{b}}{3}(\text{x}+\text{b})^\frac{3}{2}\Big]+\text{c}$ where, c is an arbitrary constant.
Hence, $\int\frac{\text{x}}{\sqrt{\text{x}+\text{a}}-\sqrt{\text{x}+\text{b}}}\text{dx}=$ $\frac{1}{\text{a}-\text{b}}\Big[\frac{2}{5}(\text{x}+\text{a})^\frac{5}{2}-\frac{2\text{a}}{3}(\text{x}+\text{a})^\frac{3}{2}+\frac{2}{5}(\text{x}+\text{b})^\frac{5}{2}-\frac{2\text{b}}{3}(\text{x}+\text{b})^\frac{3}{2}\Big]+\text{c}$ where, c is an arbitrary constant.