Evaluate the integrals in Exercises 47–68.

∫₀¹ dr .
∛(7 - 5r)²

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Identify the integral to be evaluated: \(\int_0^1 \frac{dr}{\sqrt[3]{(7 - 5r)^2}}\).

Rewrite the integrand using exponent notation: \(\frac{1}{\sqrt[3]{(7 - 5r)^2}} = (7 - 5r)^{-\frac{2}{3}}\).

Consider a substitution to simplify the integral. Let \(u = 7 - 5r\), then compute \(du\) in terms of \(dr\): \(du = -5 \, dr\), so \(dr = -\frac{1}{5} du\).

Change the limits of integration according to the substitution: when \(r=0\), \(u=7\), and when \(r=1\), \(u=7 - 5 = 2\).

Rewrite the integral in terms of \(u\) and \(du\): \(\int_7^2 u^{-\frac{2}{3}} \left(-\frac{1}{5}\right) du\). Then, reverse the limits to maintain the standard order and adjust the sign accordingly.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral calculates the net area under a curve between two limits, here from 0 to 1. It represents the accumulation of quantities and is evaluated using antiderivatives or numerical methods when an explicit antiderivative is complex.

Integration of Functions with Radicals

Integrating functions involving roots, such as cube roots, often requires rewriting the radical as a fractional exponent. This allows the use of power rule integration or substitution techniques to simplify the integral.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. For expressions like (7 - 5r), setting u = 7 - 5r helps rewrite the integral in terms of u, making it easier to integrate.

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