If , where C is the constant of integration, then is equal to __________. [2025]

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Solution

Q.
If $\int \frac{2 x^{2} + 5 x + 9}{\sqrt{x^{2} + x + 1}} d x = x \sqrt{x^{2} + x + 1} + α \sqrt{x^{2} + x + 1} + β \log_{e}$ $| x + \frac{1}{2} + \sqrt{x^{2} + x + 1} |$ $+ C$ , where C is the constant of integration, then $α + 2 β$ is equal to __________. [2025]

Ans.
(16)

Given integral is $\int \frac{2 x^{2} + 5 x + 9}{\sqrt{x^{2} + x + 1}} d x$

Using partial fraction decomposition, we get

$2 x^{2} + 5 x + 9 = A (x^{2} + x + 1) + B (2 x + 1) + C$

$= A x^{2} + (A + 2 B) x + A + B + C$

Comparing terms, we get $A = 2, B = \frac{3}{2} and C = \frac{11}{2}$

$∴ \int \frac{2 x^{2} + 5 x + 9}{\sqrt{x^{2} + x + 1}} d x = 2 \int \frac{x^{2} + x + 1}{\sqrt{x^{2} + x + 1}} d x + \frac{3}{2} \int \frac{2 x + 1}{\sqrt{x^{2} + x + 1}} d x + \frac{11}{2} \int \frac{1}{\sqrt{x^{2} + x + 1}} d x$

$= 2 \int \sqrt{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d x + \frac{3}{2} \times 2 \sqrt{x^{2} + x + 1} + \frac{11}{2} \int \frac{1}{\sqrt{{(x + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}}} d x$

$= 2 \times ((\frac{x + \frac{1}{2}}{2}) \sqrt{x^{2} + x + 1} + \frac{3}{2 \times 4} \ln | x + \frac{1}{2} + \sqrt{x^{2} + x + 1} |) + 3 \sqrt{x^{2} + x + 1} + \frac{11}{2} \ln | x + \frac{1}{2} + \sqrt{x^{2} + x + 1} | + C$

On comparing, we get $α = \frac{7}{2} and β = \frac{25}{4}$

Hence, $α + 2 β = \frac{7}{2} + 2 \times \frac{25}{4} = 16$ .

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