Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}} d x$

Solution:

Given:

$\int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}} d x$

By applying $(a+b)^{2}=a^{2}+b^{2}+2 a b$

$\Rightarrow \int \frac{(1)^{2}+(\sqrt{x})^{2}+2 \times 1 \times \sqrt{x}}{\sqrt{x}} \mathrm{dx}$

$\Rightarrow \int \frac{1+x+2 \sqrt{x}}{\sqrt{x}} d x$

By Splitting, we get,

$\Rightarrow \int\left(\frac{1}{\sqrt{x}}+\frac{x}{\sqrt{x}}+\frac{2 \sqrt{x}}{\sqrt{x}}\right) d x$

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

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

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

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

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

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