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0 votes
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in Integrals calculus by (36.9k points)
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Evaluate: ∫ (x2 – 1/ x √(x4 + 3x2 + 1)) dx

2 Answers

+1 vote
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Best answer

The given integral is 

\(\int \frac {x^2-1}{x\sqrt{x^4+3x^2 +1}}dx\)

\(=\int \frac {x^2-1}{x^2\sqrt{x^4+\frac 1{x^2} +3}}dx\)    (By taking x2 common from the root part)

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

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

Let \(x + \frac 1x = t\)

\(\therefore \left(1 - \frac 1{x^2}\right)dx = dt\)

\(= \int \frac{dt}{\sqrt{t^2 +1}}\)

\(= log |t + \sqrt{t^2 + 1}|+C\)

\(= log \left|(x + \frac 1x) + \sqrt{(x + \frac 1x)^2 + 1}\right| +C\)

\(= log \left|(x + \frac 1x) + \sqrt{x^2 + \frac 1{x^2} + 3}\right| +C\)

+2 votes
by (38.6k points)

The given integral is

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