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To integrate [tex]∫ \frac{ 1}{ \sqrt{x}(x + 1) }dx[/tex] with respect to x represent the integral as follows:

[tex]\[\int \frac{1}{\sqrt{x}(x+1)} \, dx\][/tex]

To solve this integral, we can make a substitution by letting [tex]\[\ u = \sqrt{x}\][/tex]Then, [tex]\[du/dx = \frac{1}{2\sqrt{x}}\][/tex] and [tex]\[\ dx = 2\sqrt{x} \, du\][/tex]

Now, rewrite the integral using u:

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

Simplify the integrand:

[tex]\[\int \frac{2}{u^2 + 1} \, du\][/tex]

To integrate this, we can use ;

[tex]\[\int \frac{2}{u^2 + 1} \, du = 2 tan^{-1} (u) + C_1, \quad \text{where } C_1 \text{ is the constant of integration.}\][/tex]

Now revert back to the original variable

[tex]\[\int \frac{1}{\sqrt{x}(x+1)} \, dx = 2 tan^{-1} (\sqrt{x}) + C, \quad \text{where } C = C_1 \text{ is the final constant of integration.}\][/tex]

[tex]\huge\fcolorbox{blue}{red}{\color{lime}{Ꭺɴꮪꮃꭼꭱ}}[/tex]

To integrate[tex] ∫ \frac{ 1}{ \sqrt{x}(x + 1) }[/tex]

dx with respect to x represent the integral as follows:

[tex]\int \frac{1}{\sqrt{x}(x+1)}[/tex]

To solve this integral, we can make a substitution by letting [tex]\ u = \sqrt{x}[/tex]

Then, [tex]du/dx = \frac{1}{2\sqrt{x}}[/tex]

[tex] and \ dx = 2\sqrt{x}[/tex]

Now, rewrite the integral using u:

[tex]\int \frac{1}{u(u^2 + 1)}[/tex]

[tex]\int \frac{2}{u^2 + 1}[/tex]

[tex]\int \frac{2}{u^2 + 1} \, du = 2 tan^{-1} (u) + C_1, \quad \text{where } C_1 \text{ is the constant of integration.}[/tex]

[tex]\int \frac{1}{\sqrt{x}(x+1)} \, dx = 2 tan^{-1} (\sqrt{x}) + C, \quad \text{where } C = C_1 \text{ is the final constant of integration.}[/tex]