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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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