Mathematics

Evaluate $$\int {\dfrac{{2x - 3}}{{\sqrt {{x^2} + x + 1} }}dx} $$


SOLUTION
Given that:
$$\Rightarrow\int\dfrac{2x-3}{\sqrt{x^2+x+1}} dx$$
$$\Rightarrow\int\left(\dfrac{2x+1}{\sqrt{x^2+x+1}}-\dfrac{4}{\sqrt{x^2+x+1}}\right)dx$$
$$\Rightarrow\int\dfrac{2x+1}{\sqrt{x^2+x+1}}-\int\dfrac{4}{\sqrt{x^2+x+1}}dx$$
Let $$x^2+x+1=t $$ in First part, than $$(2x+3)dx=dt$$
$$\Rightarrow\int\dfrac{1}{\sqrt t}dt-\int\dfrac{4}{\sqrt{x^2+x+\dfrac{1}{4}+\dfrac{3}{4}}}dx$$
$$\Rightarrow2\sqrt t \,-\int\dfrac{4}{\sqrt{\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}}}dx$$
$$\Rightarrow2\sqrt{x^2+x+1}-4\log\left(x+\sqrt{x^2+\dfrac{3}{4}}\right)+constant$$
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Subjective Medium Published on 17th 09, 2020
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