Mathematics

Evaluate the following : $$\displaystyle\int \dfrac{1}{4x^{2}-3}.dx$$


SOLUTION
$$I=\displaystyle\int \dfrac{1}{4x^{2}-3}.dx$$
$$=\dfrac{1}{4}\displaystyle\int \dfrac{1}{x^{2}-\dfrac{3}{4}}.dx$$
$$=\dfrac{1}{4}\displaystyle\int \dfrac{1}{x^{2}-\left(\dfrac{\sqrt{3}}{2}\right)^{2}}.dx$$
$$=\dfrac{1}{4}\dfrac{1}{2\left(\dfrac{\sqrt{3}}{2}\right)}\log \left|\dfrac{x-\dfrac{\sqrt{3}}{2}}{x+\dfrac{\sqrt{3}}{2}}\right|+c$$
$$=\dfrac{1}{4\sqrt{3}}\log \left|\dfrac{2x-\sqrt{3}}{2x+\sqrt{3}}\right|+c$$
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Subjective Medium Published on 17th 09, 2020
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