91Ó°ÊÓ

Differentiation is a mathematical process that computes the rate at which a function is changing at any given point. It is one of the key concepts in calculus. Differentiation allows us to find derivatives, which tell us how a function's value changes in response to changes in input. In the context of our exercise, differentiation is used to verify the given integral identity. We do this by differentiating the right side of the expression to see if it matches the integrand on the left side. This involves applying rules such as the product rule, which we'll discuss next. Differentiation is fundamental because it connects various areas of calculus, facilitating integration and understanding of functions' behavior.

The product rule is a specific rule of differentiation used when a function is the product of two other functions. It states that the derivative of the product of two functions, say, \( u(x) \) and \( v(x) \), is given by:\[ (uv)' = u'v + uv' \]This rule is essential because it enables us to handle situations where functions are multiplied, such as in the expression \( \sqrt{x-1}(3x^2 + 4x + 8) \). In our exercise, the product rule is employed to differentiate this expression—as the product of \( u = \sqrt{x-1} \) and \( v = 3x^2 + 4x + 8 \). Calculating the derivatives of \( u \) and \( v \) and substituting them into the product rule formula allows us to examine how the overall function changes, helping us verify the integral's validity.

An integral identity is an equation that demonstrates the equality of two expressions by integrating both sides. The principle here is that when differentiated, both sides of the equation should yield the same function, confirming the identity.In this exercise, our goal is to prove the identity:\[ \int \frac{x^2}{\sqrt{x-1}} \, dx = \frac{2}{15} \sqrt{x-1}(3x^2 + 4x + 8) + C \]To verify this, we differentiate the right side and compare it to the integrand \( \frac{x^2}{\sqrt{x-1}} \) on the left side. If both are identical, the integral identity is validated. Integral identities are crucial as they enable functional transformations and simplifications, particularly in calculus, where identifying similar expressions helps in evaluating integrals and finding solutions.

The substitution method is a popular technique used to simplify integration problems by making a suitable substitution that transforms the integral into a more manageable form. It often involves substituting expressions with a new variable to rewrite the integral.In our case, we use the substitution \( u = x-1 \). This choice changes the variable from \( x \) to \( u \), making the integral easier to solve by converting it into:\[ \int \frac{(u+1)^2}{\sqrt{u}} \, du \]The main goal is to identify a simpler function \( f(u) \) that the new integral represents. By expanding and simplifying \( (u+1)^2 \), we rewrite the original complex integrand in terms of powers of \( u \), ultimately giving us \( f(u) = u^{1/2} + 2u^{-1/2} + u^{-3/2} \). This method streamlines the process of integration, turning difficult problems into basic ones.