Final answer:
To solve the integration problem using trigonometric substitution, re-express the integrand for a suitable trigonometric identity, substitute, integrate, and revert the substitution to find the integral in terms of x.
Step-by-step explanation:
To integrate the expression x divided by the square root of the quantity x squared minus 8x plus 25 from 4 to 7 with respect to x, using trigonometric substitution, we first need to manipulate the integrand into a form suitable for the substitution. We identify the expression under the square root as a perfect square trinomial which can be written as (x - 4)2 + 9. A common trigonometric substitution for an integrand that contains a 2 + b2 term is x = a sec(θ) or x - 4 = 3 sec(θ), which simplifies the square root to 3 tan(θ).
After substituting, simplifying, and finding the bounds in terms of θ, we can then integrate with respect to θ and, lastly, convert back to x using the inverse trigonometric functions.
This type of problem is a classic example of using trigonometric substitution for integration, especially when dealing with a radical expression that resembles the Pythagorean theorem, and is a common technique used in calculus.