Write the primitive or anti-derivative of f(x)= x =1 x . - Vidyadip

Question

Write the primitive or anti-derivative of $\text{f(x)}=\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}$.

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Answer

$\text{f(x)}=\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}$
Integrating both sides:
$\int\text{f(x)}\text{dx}=\int\Big(\sqrt{\text{x}}=\frac{1}{\sqrt{\text{x}}}\Big)\text{dx}$
$=\int\Big(\text{x}^{\frac{1}{2}}+\text{x}^{-\frac{1}{2}}\Big)\text{dx}$
$=\bigg[\frac{\text{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\bigg]+\bigg[\frac{\text{x}^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\bigg]+\text{C}$
$=\frac{2}{3}\text{x}^{\frac{3}{2}}+2\text{x}^{\frac{1}{2}}+\text{C}$

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