∫(1 to 10) 2624 / (x² + x + 1) dx

Explanation:

Step1: Complete the square for the denominator

First, complete the square for $x^{2}+x + 1$. We have $x^{2}+x + 1=(x+\frac{1}{2})^{2}+\frac{3}{4}$. Let $u=x+\frac{1}{2}$, then $du = dx$. When $x = 1$, $u=\frac{3}{2}$; when $x = 10$, $u=\frac{21}{2}$. The integral becomes $\int_{\frac{3}{2}}^{\frac{21}{2}}\frac{2624}{u^{2}+\frac{3}{4}}du$.

Step2: Use the integral formula $\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$

We can rewrite $\int_{\frac{3}{2}}^{\frac{21}{2}}\frac{2624}{u^{2}+\frac{3}{4}}du$ as $2624\int_{\frac{3}{2}}^{\frac{21}{2}}\frac{1}{u^{2}+(\frac{\sqrt{3}}{2})^{2}}du$. According to the formula $\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$ with $a = \frac{\sqrt{3}}{2}$, we get $2624\times\frac{2}{\sqrt{3}}\left[\arctan\left(\frac{2u}{\sqrt{3}}\right)\right]_{\frac{3}{2}}^{\frac{21}{2}}$.

Step3: Evaluate the definite - integral

$2624\times\frac{2}{\sqrt{3}}\left(\arctan\left(\frac{2\times\frac{21}{2}}{\sqrt{3}}\right)-\arctan\left(\frac{2\times\frac{3}{2}}{\sqrt{3}}\right)\right)= \frac{5248}{\sqrt{3}}\left(\arctan(7\sqrt{3})-\arctan(\sqrt{3})\right)$. Using the formula $\arctan A-\arctan B=\arctan\left(\frac{A - B}{1+AB}\right)$, we have $\frac{5248}{\sqrt{3}}\arctan\left(\frac{7\sqrt{3}-\sqrt{3}}{1 + 7\sqrt{3}\times\sqrt{3}}\right)=\frac{5248}{\sqrt{3}}\arctan\left(\frac{6\sqrt{3}}{1 + 21}\right)=\frac{5248}{\sqrt{3}}\arctan\left(\frac{6\sqrt{3}}{22}\right)=\frac{5248}{\sqrt{3}}\arctan\left(\frac{3\sqrt{3}}{11}\right)$.

Answer:

$\frac{5248}{\sqrt{3}}\arctan\left(\frac{3\sqrt{3}}{11}\right)$