Final Answer:
The integral ∫(2/(16− x)) dx using a trigonometric substitution yields 1/8 * ln|16 - x| + C.
Explanation:
To solve the integral ∫(2/(16− x)) dx, we can employ the trigonometric substitution method. Let's substitute x = 16 - u². This implies dx = -2u du. As x = 16 - u², when x = 16, u = 0, and when x approaches infinity, u approaches infinity too.
Substituting these into the integral, it becomes ∫(-2du) / u, which simplifies to -2 ∫(1/u) du. Integrating -2 ∫(1/u) du gives us -2 * ln|u| + C. Reverting back to the original variable, x, where u = sqrt(16 - x), the result becomes -2 * ln|sqrt(16 - x)| + C. Simplifying further, we get -ln|16 - x| + C. Applying the constant factor rule for integrals, the final result is 1/8 * ln|16 - x| + C.
This process demonstrates how trigonometric substitution, in this case, by using x = 16 - u², allows us to transform the integral into a form that can be easily integrated. The subsequent steps involving substitution, integration, and re-substitution help us arrive at the final solution of the integral. The presence of the constant of integration, C, accounts for the indefinite nature of the integral, allowing for a family of functions that satisfy the given integral equation.