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\(\int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)}\) equals ______(a) \(\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{5} tan^{-1}x+5C\)(b) \(\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{6} tan^{-1}x+C\)(c) \(\frac{3}{5}log|x+2| + \frac{1}{6}log|x^2+1|+\frac{1}{6} tan^{-1}x+C\)(d) \(\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{5} tan^{-1}x+C\)This question was addressed to me during an online interview.My doubt stems from Integration by Partial Fractions topic in division Integrals of Mathematics – Class 12

Answer»

Correct option is (d) \(\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{5} tan^{-1}x+C\)

Explanation: \(\INT \frac{(x^2+x+1)dx}{(x+2)(x^2+1)} = \frac{A}{(x+2)} + \frac{Bx+C}{(x^2+1)}\)

Now equating, (x^2+x+1) = A (x^2+1) + (Bx+C) (x+2)

After equating and SOLVING for coefficient we get values,

A=\(\frac{3}{5}\), B=\(\frac{2}{5}\), C=\(\frac{1}{5}\), now putting these values in the EQUATION we get,

\(\int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)} = \frac{3}{5} \int \frac{dx}{(x+2)} + \frac{1}{5} \int \frac{2xdx}{(x^2+1)} + \frac{1}{5} \int \frac{dx}{(x^2+1)}\)

Hence it COMES, \(\frac{3}{5} log|x+2| + \frac{1}{5} log|x^2+1|+\frac{1}{5}tan^{-1}x+C\)


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