91Ó°ÊÓ

The substitution method is a powerful integration technique used to simplify complex integrals. It involves replacing a part of the integrand with a new variable to make the integral easier to evaluate. In this exercise, we initially have the integral \(\int x \sqrt{2-x} \, dx\)\. To use substitution, we set up a change of variable, which typically simplifies the integrand considerably.\Steps in the substitution process:\

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• Identify a suitable substitution. Here, we let \(\u = 2 - x\) because the square root function will become simpler under this new variable.\
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• Compute the differential of the substitution. For \(\u = 2 - x\), we have \(\du = -dx\) or equivalently \(\dx = -du\).
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• Substitute \(\u\) and \(\dx\) into the integral, converting all occurrences of the original variable (here, \(\x\)) into terms of the new variable (\(\u\)).\
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• Simplify and evaluate the integral in terms of the new variable.\
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• Finally, back-substitute the original variable into the solution to express the answer in terms of the original variable.\
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\With our problem, the integral \(\int x \sqrt{2-x} \, dx\) becomes easier to handle once we swap \(\x\) and \(\sqrt{2-x}\) for expressions involving \(\u\).

An indefinite integral represents a family of functions whose derivative is the integrand. Unlike definite integrals, which compute the area under a curve between two points, indefinite integrals include a constant of integration, denoted by \(\text{C}\). In this example, the indefinite integral of the function \(\int x \sqrt{2-x} \, dx\) involves finding an antiderivative without specified limits of integration.\Key Points:\

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• The solution includes an arbitrary constant \(\text{C}\) because there are infinitely many functions differing by a constant that can satisfy the integral equation.\
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• The notation for indefinite integrals is \(\int f(x) \, dx = F(x) + C\), where \(\text{F} (x)\) is an antiderivative of \(\text{f} (x)\).\
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\In this exercise, after performing the substitution and integrating, we obtain the antiderivatives relative to \(\u\) and then convert back, resulting in \(\text{-} \frac{\text{4}}{\text{3}} (2 - x)^{3/2} + \frac{\text{2}}{\text{5}} (2 - x)^{5/2} + C\). This encapsulates all functions that could be derivatives of \(\text{x} \sqrt{2-x}\).

Integrating powers of \(\text{u}\) is a straightforward process and an essential aspect of substitution, particularly when the integrand is expressed in terms of a polynomial. Once the integrand is transformed to a function of \(\text{u}\), we can apply basic power rules of integration. For this problem, after substitution, the integral becomes a sum of terms of \(\text{u}\) raised to various powers.\Basic Integration Rule:\

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• The integral of \(\text{u}^n\) is given by \(\frac{u^{n+1}}{n+1}\) provided \(\text{n} eq \text{-1}\).\
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\First, we expanded \((2-u) \sqrt{u}\) to get \(\text{2u}^\frac{\text{1}}{\text{2}} - \text{u}^\frac{\text{3}}{\text{2}}\). We then integrated each term separately: \(\text{2u}^\frac{\text{1}}{\text{2}}\) becomes \(\frac{\text{4}}{\text{3}}\u^{\frac{\text{3}}{\text{2}}}\) and \(\text{u}^\frac{\text{3}}{\text{2}}\) becomes \(\frac{\text{2}}{\text{5}} \u^{\frac{\text{5}}{\text{2}}}\).\Putting it all together:\

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• Combine the integrated terms, taking care of constants and signs. Here, we get: \(\text{-}\frac{\text{4}}{\text{3}} \u^{\frac{\text{3}}{\text{2}}} + \frac{\text{2}}{\text{5}} \u^{\frac{\text{5}}{\text{2}}}\).
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• Always add the constant of integration \(\text{C}\).
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\Finally, to express our answer in terms of \(\text{x}\), we back-substitute \(\text{u} = 2 - x\).