Mathematics

Solve $$\displaystyle\int { \frac { 1 }{ \sqrt { 3x+4 } -\sqrt { 3x+1 }  } dx }$$ 


SOLUTION
$$=\displaystyle \int \dfrac{1}{\sqrt{3x+4}-\sqrt{3x+1}}dx$$    Rationlize

$$=\displaystyle \int \dfrac{\sqrt{3x+4}\sqrt{3x+1}}{3x+4-3x-1}dx$$

$$=\displaystyle \dfrac{1}{3}\int (\sqrt{3x+4}+\sqrt{3x+1})dx$$

$$=\displaystyle \dfrac{1}{3}\left [ \dfrac{(3x+4)^{3/2}}{\dfrac{3}{2}\times 3} +\dfrac{(3x+1)^{3/2}}{3\times 3/2}\right ]+C$$

$$=\displaystyle \dfrac{1}{3}\left [ \dfrac{2}{9}(3x+4)^{3/2} +\dfrac{2}{9}(3x+1)^{3/2}\right ]+C$$
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Subjective Medium Published on 17th 09, 2020
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