sympy import symbols, sec, cos, integrate, pi

Define the variable and the function to integrate

x = symbols('x') function = -2 * sec(x)**2 - 5 * cos(x)

Define the integration limits

lower_limit = pi/4 upper_limit = pi/3

Perform the definite integration

result = integrate(function, (x, lower_limit, upper_limit)) resultThe result of the definite integral

$\int_{\pi/4}^{\pi/3} \left(-2 \sec^2(x) - 5 \cos(x)\right) dx$

is:

$-\frac{9\sqrt{3}}{2} + 2 + \frac{5\sqrt{2}}{2}.$

Would you like me to explain this step-by-step or clarify any part?

Here are five related questions to deepen your understanding:

1. How do you determine whether a function is integrable over a given interval?
2. What are the basic rules for integrating trigonometric functions like $\sec^2(x)$ and $\cos(x)$?
3. How does the definite integral differ from an indefinite integral conceptually?
4. Why are limits of integration important in solving definite integrals?
5. What applications in physics or engineering might involve similar integrals?

Tip: When working with trigonometric integrals, recalling standard derivatives and antiderivatives of trigonometric functions is crucial for efficient computation.