Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{1}{\sqrt{x}}\left(1+\frac{1}{x}\right) d x$

Solution:

Given:

$\int \frac{1}{\sqrt{x}}\left\{1+\frac{1}{x}\right\} d x$

By multiplying $\frac{1}{\sqrt{x}}$ with inside brackets,

$\Rightarrow \int\left\{\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x}} \times \frac{1}{x}\right\} d x$

$\Rightarrow \int\left\{\frac{1}{x^{\frac{1}{2}}}+\frac{1}{x^{\frac{1}{2}}} \times \frac{1}{x}\right\} d x$

$\Rightarrow \int\left\{\frac{1}{x^{\frac{1}{2}}}+\frac{1}{x^{\frac{1}{2}+1}}\right\} d x$

$\Rightarrow \int\left\{\frac{1}{x^{\frac{1}{2}}}+\frac{1}{x^{\frac{3}{2}}}\right\} d x$

By Splitting them, we get,

$\Rightarrow \int x^{-\frac{1}{2}} d x+\int x^{-\frac{3}{2}} d x$

By applying the formula,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\frac{x^{-\frac{3}{2}+1}}{-\frac{3}{2}+1}+c$

$\Rightarrow \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+\frac{x^{-\frac{1}{2}}}{-\frac{1}{2}}+c$

$\Rightarrow 2 x^{\frac{1}{2}}-2 x^{-\frac{1}{2}}+c$

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