solve to get a code: \n∫¹⁰₁ 2624 / (x² + x + 1) dx\nask other players\ndiscord community

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 the integral as $2624\int_{\frac{3}{2}}^{\frac{21}{2}}\frac{1}{u^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}du$. According to the formula $\int\frac{1}{a^{2}+x^{2}}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$ (here $a = \frac{\sqrt{3}}{2}$), 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

[ \begin{align*} &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)\ =&\frac{5248}{\sqrt{3}}\left(\arctan\left(\frac{7\sqrt{3}-\sqrt{3}}{1 + 7\sqrt{3}\times\sqrt{3}}\right)\right)\ =&\frac{5248}{\sqrt{3}}\left(\arctan\left(\frac{6\sqrt{3}}{1 + 21}\right)\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) \end{align*} ]

Answer:

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