Final answer:
The question asks to evaluate two mathematical integrals and discusses work done by a variable force over a displacement. Without specific limits or further details, only the methods, such as trigonometric substitutions or partial fractions, which could simplify these integrals, are suggested.
Step-by-step explanation:
To evaluate the given integral ∫cosθ dθ/sin ²θ-6 sinθ-8, we would need to suggest methods such as substitution or partial fractions to simplify the integral. However, without further context or detailed information provided in the question, it's not possible to give a complete step-by-step evaluation. As for the second portion of the question concerning ∫√(1+x²)/x dx, we would typically use a trigonometric substitution like x = tan(θ) to simplify the integral under a radical. Again, without the limits of integration being specified, we can only discuss the methods to tackle the integral rather than evaluate it explicitly.
To find the work done by a force F(x) = (10 N)sin[(0.1 m⁻¹)x] from x = 0 to x = 10 m, we would perform the integral of F(x) dx over the given limits. This mechanical work integral would result in the total work done by the variable force over the displacement.