Step 1: Rewrite the denominator using a standard formula
The integral involves \(\sin(x)\), so we aim to simplify using trigonometric identities. The standard substitution is to express the integral in terms of the tangent half-angle substitution:
\(\begin{aligned} \tan\left(\frac{x}{2}\right) = t, \quad \sin(x) = \frac{2t}{1 + t^2}, \quad \cos(x) = \frac{1 - t^2}{1 + t^2}, \quad dx = \frac{2 \, dt}{1 + t^2} \end{aligned}\).
Step 2: Substitute into the integral
Using the substitution, the denominator becomes:
\(\begin{aligned} 2 + 5\sin(x) = 2 + 5\left(\frac{2t}{1 + t^2}\right) = \frac{2(1 + t^2) + 10t}{1 + t^2} = \frac{2 + 2t^2 + 10t}{1 + t^2} \end{aligned}\)
The integral becomes:
\(\begin{aligned} I = \int \frac{1 + t^2}{2 + 2t^2 + 10t} \cdot \frac{2 \, dt}{1 + t^2} \end{aligned}\)
Simplify:
\(\begin{aligned} I = \int \frac{2 \, dt}{2 + 2t^2 + 10t} \end{aligned}\)
Step 3: Simplify the denominator
Factorize the denominator \(2 + 2t^2 + 10t\):
\(2 + 2t^2 + 10t = 2(t^2 + 5t + 1)\)
So the integral becomes:
\(\begin{aligned} I = \int \frac{dt}{t^2 + 5t + 1} \end{aligned}\)
Step 4: Complete the square
Complete the square for \(t^2 + 5t + 1\):
\(\begin{aligned} t^2 + 5t + 1 = \left(t + \frac{5}{2}\right)^2 - \frac{21}{4} \end{aligned}\)
Thus, the integral becomes:
\(\begin{aligned} I = \int \frac{dt}{\left(t + \frac{5}{2}\right)^2 - \left(\frac{\sqrt{21}}{2}\right)^2} \end{aligned}\)
Step 5: Use the standard form of the integral
This is now in the standard form:
\(\begin{aligned} \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \ln\left|\frac{x - a}{x + a}\right| + C \end{aligned}\)
Let \(\begin{aligned} u = t + \frac{5}{2} \end{aligned}\), so the integral becomes:
\(\begin{aligned} I = \frac{1}{\sqrt{21}} \ln\left|\frac{u - \frac{\sqrt{21}}{2}}{u + \frac{\sqrt{21}}{2}}\right| + C \end{aligned}\)
Substituting back \(\begin{aligned} u = t + \frac{5}{2} \end{aligned}\) and \(\begin{aligned} t = \tan\left(\frac{x}{2}\right) \end{aligned}\), the final result is:
\(\begin{aligned} I = \frac{1}{\sqrt{21}} \ln\left|\frac{\tan\left(\frac{x}{2}\right) + \frac{5}{2} - \frac{\sqrt{21}}{2}}{\tan\left(\frac{x}{2}\right) + \frac{5}{2} + \frac{\sqrt{21}}{2}}\right| + C \end{aligned}\)
