Substitution is used to simplify the integrand and make it easier to solve. In this case, substituting u = 16-x will eliminate the square root and make the integral solvable using basic integration techniques.
The most effective substitution for this integral is u = 16-x, as it eliminates the square root and simplifies the integrand. Generally, the best substitution is one that simplifies the integrand and makes it easier to solve.
The steps for solving this integral are as follows:
1. Identify the appropriate substitution, in this case u = 16-x
2. Rewrite the integrand in terms of u
3. Substitute u back into the integral
4. Solve the resulting integral using basic integration techniques
5. Substitute back in the original variable, x, to get the final answer.
Yes, this integral can also be solved using integration by parts. However, substitution is the more efficient and straightforward method in this case.
No, substitution can only be used for certain types of integrals where the substitution simplifies the integrand. For example, it would not be effective for integrals involving trigonometric functions.