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\int\frac{3x-1}{2x^2+x+1}dx

Integral of (3x-1)/(x+1+2x^2)

Answer

$\frac{1}{2}\left(3\left(\frac{1}{2}\ln\left|\frac{1}{2}x+x^2+\frac{1}{2}\right|-\frac{2}{\sqrt{4}}arctan\left(\sqrt{2}\left(\frac{1}{4}+x\right)\right)\right)-\sqrt{2}arctan\left(\sqrt{2}\left(\frac{1}{4}+x\right)\right)\right)+C_0$

Step-by-step explanation

Problem

$\int\frac{3x-1}{2x^2+x+1}dx$
1

Use the complete the square method to factor the trinomial of the form $ax^2+bx+c$. Take common factor $a$ ($2$) to all terms

$\int\frac{3x-1}{2\left(\frac{1}{2}+\frac{1}{2}x+x^2\right)}dx$

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Answer

$\frac{1}{2}\left(3\left(\frac{1}{2}\ln\left|\frac{1}{2}x+x^2+\frac{1}{2}\right|-\frac{2}{\sqrt{4}}arctan\left(\sqrt{2}\left(\frac{1}{4}+x\right)\right)\right)-\sqrt{2}arctan\left(\sqrt{2}\left(\frac{1}{4}+x\right)\right)\right)+C_0$

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