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Integrate the function
$\frac{4 x+1}{\sqrt{2 x^2+x-3}}$
$\frac{4 x+1}{\sqrt{2 x^2+x-3}}$
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Verified Answer
Let $\mathrm{I}=\int \frac{4 x+1}{\sqrt{2 x^2+x-3}} d x$
Put $2 x^2+x-3=t$, so that $(4 x+1) d x=d t$
$\therefore \quad \mathrm{I}=\int \frac{d t}{\sqrt{t}}=2 t^{\frac{1}{2}}+\mathrm{C}=2 \sqrt{2 x^2+x-3}+\mathrm{C} .$
Put $2 x^2+x-3=t$, so that $(4 x+1) d x=d t$
$\therefore \quad \mathrm{I}=\int \frac{d t}{\sqrt{t}}=2 t^{\frac{1}{2}}+\mathrm{C}=2 \sqrt{2 x^2+x-3}+\mathrm{C} .$
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