If ∫x3+x+1x4+x2+1 dx=12f(x)+13g(x)+C, C is constant of integration. Given f(0)=0, g(1)=π6 then which of the following is/are TRUE?

∫x3+x+1x4+x2+1 dx=12∫2x3+2x+2x4+x2+1 dx =12[∫2x+1x2+x+1dx+∫1x2−x+1dx] =12log|x2+x+1|+13tan−1(2x+13)+13tan−1(x-1x3)C

If ∫x3+x+1x4+x2+1 dx=12f(x)+13g(x)+C, C is constant of integration. Given f(0)=0, g(1)=π6 then which of the following is/are TRUE?

∫x3+x+1x4+x2+1 dx=12∫2x3+2x+2x4+x2+1 dx =12[∫2x+1x2+x+1dx+∫1x2−x+1dx] =12log|x2+x+1|+13tan−1(2x+13)+13tan−1(x-1x3)C

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If $\int \frac{x^{3} + x + 1}{x^{4} + x^{2} + 1} d x = \frac{1}{2} f (x) + \frac{1}{\sqrt{3}} g (x) + C$ , C is constant of integration. Given $f (0) = 0, g (1) = \frac{π}{6}$ then which of the following is/are TRUE?

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$f (1) = \ln 2$

$f (2) = \ln 7$

$g (2) = \frac{π}{3}$

$g (- 2) = \frac{- π}{3}$

answer is B.

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Detailed Solution

$\int \frac{x^{3} + x + 1}{x^{4} + x^{2} + 1} d x = \frac{1}{2} \int \frac{2 x^{3} + 2 x + 2}{x^{4} + x^{2} + 1} d x = \frac{1}{2} [\int \frac{2 x + 1}{x^{2} + x + 1} d x + \int \frac{1}{x^{2} - x + 1} d x] = \frac{1}{2} \log | x^{2} + x + 1 | + \frac{1}{\sqrt{3}} \tan^{- 1} (\frac{2 x + 1}{\sqrt{3}}) + \frac{1}{\sqrt{3}} \tan^{- 1} (\frac{x - \frac{1}{x}}{\sqrt{3}}) C$