If $\int\frac{2x^2+5x+9}{\sqrt{x^2+x+1}}dx=x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\left|x+\frac12+\sqrt{x^2+x+1}\right|+C$, where $C$ is the constant of integration, then $\alpha+2\beta$ is equal to ________.

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Differentiate the right side with $Q=x^2+x+1$: $\frac{4x^2+3x+2+\alpha(2x+1)+2\beta}{2\sqrt Q}$. Equating with $\frac{2x^2+5x+9}{\sqrt Q}$: $3+2\alpha=10\Rightarrow\alpha=\frac72$ and $2+\alpha+2\beta=18\Rightarrow\beta=\frac{25}4$. $\alpha+2\beta=\frac72+\frac{25}2=16$. Answer: 16.

Q. If $\int\frac{2x^2+5x+9}{\sqrt{x^2+x+1}}dx=x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\left|x+\frac12+\sqrt{x^2+x+1}\right|+C$, where $C$ is the constant of integration, then $\alpha+2\beta$ is equal to ________.

Explanation

Reviewed by Dr. Abubacker

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