The given integral is:

$\int_{1}^{10} \frac{2624}{x^2 + x + 1} \, dx$

Let me solve this step by step.

Step 1: Simplify the integral

The general integral form for $\frac{1}{x^2 + x + 1}$ is often approached by completing the square for the quadratic denominator. Completing the square for $x^2 + x + 1$: $x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4}$

Thus, the denominator becomes: $x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2$

This is now in the form of $a^2 + u^2$, which suggests using the standard arctangent integration formula.

Step 2: Apply the formula for arctangent

The integral of $\frac{1}{a^2 + u^2}$ is: $\int \frac{1}{a^2 + u^2} \, du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$

Here:

• $u = x + \frac{1}{2}$,
• $du = dx$,
• $a = \frac{\sqrt{3}}{2}$.

Substitute these into the integral: $\int \frac{1}{x^2 + x + 1} \, dx = \frac{2}{\sqrt{3}} \arctan\left(\frac{2(x + \frac{1}{2})}{\sqrt{3}}\right) + C$

Step 3: Multiply by the constant 2624

The integral becomes: $\int_{1}^{10} \frac{2624}{x^2 + x + 1} \, dx = 2624 \cdot \frac{2}{\sqrt{3}} \left[ \arctan\left(\frac{2(x + \frac{1}{2})}{\sqrt{3}}\right) \right]_1^{10}$

Simplify the constant: $2624 \cdot \frac{2}{\sqrt{3}} = \frac{5248}{\sqrt{3}}$

Step 4: Evaluate the definite integral

The antiderivative is: $\frac{5248}{\sqrt{3}} \left[ \arctan\left(\frac{2(x + \frac{1}{2})}{\sqrt{3}}\right) \right]_1^{10}$

Substitute the bounds:

1. At $x = 10$: $\arctan\left(\frac{2(10 + \frac{1}{2})}{\sqrt{3}}\right) = \arctan\left(\frac{2 \cdot 10.5}{\sqrt{3}}\right) = \arctan\left(\frac{21}{\sqrt{3}}\right)$

2. At $x = 1$: $\arctan\left(\frac{2(1 + \frac{1}{2})}{\sqrt{3}}\right) = \arctan\left(\frac{2 \cdot 1.5}{\sqrt{3}}\right) = \arctan\left(\frac{3}{\sqrt{3}}\right) = \arctan(\sqrt{3})$

Thus, the result is: $\frac{5248}{\sqrt{3}} \left[ \arctan\left(\frac{21}{\sqrt{3}}\right) - \arctan(\sqrt{3}) \right]$

Step 5: Simplify the expression

You can evaluate the arctan values numerically or leave them in terms of their expressions for further computation. Let me know if you'd like me to calculate this final expression numerically or simplify further.

Would you like a detailed numerical evaluation? Let me know! 😊

Here are related questions to expand your understanding:

1. What is the general method for solving integrals with irreducible quadratic denominators?
2. How does completing the square help in integration problems like this?
3. What are the trigonometric forms used in integral evaluation?
4. How do we compute definite integrals involving arctangent?
5. Can partial fractions be used for other forms of quadratic denominators?

Tip: For integrals with quadratic expressions, always check if completing the square simplifies the denominator!