Mathematics

$$\int { \sqrt { 3{ x }^{ 2 }-4x+5 }  } dx$$


SOLUTION

We have,

$$ \int{\sqrt{3{{x}^{2}}-4x+5}dx} $$

$$ \Rightarrow \int{\sqrt{3\left( {{x}^{2}}-\dfrac{4}{3}x+\dfrac{5}{3} \right)}dx} $$

$$ \Rightarrow \int{\sqrt{3}}\sqrt{\left( {{x}^{2}}-\dfrac{4}{3}x+\dfrac{5}{3} \right)}dx $$

On adding and subtracting coefficient of x

Then,

$${{\left( \dfrac{4}{6} \right)}^{2}}=\dfrac{4}{9}$$

Then,

$$ \int{\sqrt{3}}\sqrt{{{x}^{2}}-\dfrac{4}{3}x+\dfrac{4}{9}-\dfrac{4}{9}+\dfrac{5}{3}}dx $$

$$ \Rightarrow \int{\sqrt{3}}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-\dfrac{11}{9}}dx $$

$$ \Rightarrow \int{\sqrt{3}}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}dx $$

On applying

$$\int{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\dfrac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}-\dfrac{{{a}^{2}}}{2}\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|$$

$$\int{\sqrt{3}}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}dx=\dfrac{\left( x-\dfrac{2}{3} \right)}{2}\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}-\dfrac{{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}}{2}\ln \left| \left( x-\dfrac{2}{3} \right)+\sqrt{{{\left( x-\dfrac{2}{3} \right)}^{2}}-{{\left( \dfrac{\sqrt{11}}{3} \right)}^{2}}} \right|$$

Hence, this is the answer.
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