Evaluate Integral with Power Function

Evaluate the integral $\displaystyle \int_{1}^{4} \frac{1}{(4-x)^{2/3}} \, dx$.

In this problem, we encounter an integral that involves a power function in the denominator. Evaluating such an integral requires a strategic approach to integration techniques. Power functions often call for algebraic manipulation or substitution techniques to simplify the expression before integrating. For integrals with the form presented in this problem, you might consider how altering the exponent or employing a substitution could aid in the simplification process. Substitution can transform the integral into a more familiar form, making it easier to evaluate. If considering substitution, observe how changing the variable affects the limits of integration and ensure these are properly updated in the integral expression itself.

In the broader context of calculus, this type of problem forms part of a larger strategy for evaluating more complex integrals. It involves understanding both the function's behavior in the specified limits and how the function's structure can dictate the most efficient method for integration. As such, this problem exemplifies the importance of flexibility in mathematical tactics—it showcases how distinct integration strategies, such as substitution, are sometimes required beyond simple antiderivative application, allowing for successful evaluation of the definite integral.

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