91Ó°ÊÓ

To tackle complex integrals, the substitution method is a handy tool that simplifies the integration process. The key idea behind substitution is to transform the original variable into a new one, making the integral easier to solve. In the case of the given problem, \(u = \sqrt{x}\) is chosen as the substitution. This transforms \(x\) into \(u^2\), which simplifies to: \(x = u^2\). The differential \(dx\) is converted using the derivative of \(u\), resulting in \(dx = 2u \, du\). Applying these changes to the integral results in:

• \(\int \sqrt{x} \sin(\sqrt{x}) \, dx = 2 \int u^2 \sin(u) \, du\)

Substitution allows the integral to morph into a simpler form, where standard techniques like integration by parts become more manageable. With the substitutions complete, the problem transitions to the integration by parts method next.

Integration by parts is a powerful technique used to integrate products of functions. Based on the product rule for differentiation, it's perfect for integrals involving products like \(u^2 \sin u\), seen in the transformed integral from substitution. The integration by parts formula is:\[\int u \, dv = uv - \int v \, du\]In this problem, we choose:

• \(u = u^2\) with a differential of \(du = 2u \, du\)
• \(dv = \sin u \, du\), resulting in \(v = -\cos u\)

Using these, the integration by parts formula becomes\[\int u^2 \sin u \, du = -u^2 \cos u + \int (-2u \cos u) \, du\]This integral suggests the need for another round of integration by parts to tackle \(\int 2u \cos u \, du\), with new choices such as \(u = 2u\) and \(dv = \cos u \, du\). After solving this, the parts are combined into a single expression ready for back substitution.

Understanding the distinction between definite and indefinite integrals is crucial in calculus. Both have unique roles:- **Indefinite Integrals**: These represent the family of all antiderivatives of a function. They are expressed with a constant \(C\), since any antiderivative plus a constant gives another antiderivative. The solution to our problem, \(-x \cos(\sqrt{x}) + 2 \sqrt{x} \sin(\sqrt{x}) + 2 \cos(\sqrt{x}) + C\), is an indefinite integral, noting the constant \(C\). - **Definite Integrals**: These compute the net area under a curve between specific limits. It results in a numerical value rather than an algebraic expression. Applying these concepts, the integral in the problem is indefinite, as there are no specified bounds. This allows the solution to express a general antiderivative of the integrand. Adding the constant \(C\) acknowledges the entire set of possible solutions that differ by a constant.