2x^2+5x+9x^2+x+1,dx = xx^2+x+1+x^2+x+1+_e|x+12+x^2+x+1|+C, where C is the constant of inte…

Question

$\displaystyle\int\frac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx = x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\left|x+\dfrac12+\sqrt{x^2+x+1}\right|+C$, where $C$ is the constant of integration. Then $\alpha+2\beta$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 16

Accepted Answer

Write $2x^2+5x+9 = A(x^2+x+1)+B(2x+1)+D$ (comparing coefficients: $A=2,\,B=\tfrac32,\,D=\tfrac{11}2$):

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

Evaluating the standard forms gives $\alpha = \dfrac72$ and matching the $\log$ coefficient, $\beta$, so that

$$\alpha+2\beta = 16$$

Solutions · 1