91Ó°ÊÓ

Integration by substitution is a technique used to evaluate integrals that are otherwise difficult to integrate directly. Think of it as the reverse process of the chain rule in differentiation. In our exercise, the integral

\[ \int 5 u \sqrt[3]{1-u^{2}} d u \]

becomes more approachable when we choose a substitution to simplify the integrand. By setting

\(v = 1 - u^{2}\),

we transform the integral into a new variable, making it easier to work with. We calculated

\(d v = -2 u d u\),

and therefore

\(-1/2 dv = u du\).

Then we substitute both

\(u du\)

and

\(1-u^{2}\) (\(v\))

in the original integral to rewrite it in terms of \(v\).This technique is invaluable as it transforms complex integrals into simpler forms that are easier to evaluate.

The power rule of integration is a fundamental concept that allows us to integrate functions of the form

\(x^n\).

To apply the power rule, we increase the exponent by one and divide by the new exponent. For instance, the indefinite integral of

\(x^n\)

is

\(\frac{x^{n+1}}{n+1} + C\),

where \(C\) is the constant of integration. In the given problem, after applying the integration by substitution, we obtained

\[-5/2 \int v^{1/3} d v\],

which requires the power rule for integration. By adding 1 to the exponent and dividing by the new exponent, the integral evaluates to

\[-15/8 v^{4/3}\].

The power rule simplifies the process of finding integrals, making it an essential tool for solving integration problems.

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables. For the purpose of checking our work in integration, differentiation acts as our quality control. By taking the derivative of our answer and comparing it to the original function, we can confirm the correctness of our integral. In our exercise example, after finding the indefinite integral

\[-15/8 (1 - u^{2})^{4/3}\],

we differentiate this expression with respect to \(u\) to verify our solution. The process of differentiation should reverse integration, leading us back to the integrand of

\[5 u \sqrt[3]{1-u^{2}}\].

If successful, differentiation not only gives us confidence in our work but also deepens our understanding of how integration and differentiation are interconnected operations.