91Ó°ÊÓ

Substitution in integration is a powerful technique. It simplifies complex integrals by changing variables. In this problem, we aim to evaluate \( \int x \sin \left(x^{2}\right) dx \). The trick is to choose a substitution that makes the problem easier to solve. Here, we let \( u = x^{2} \). This choice is interesting because the derivative of \( x^{2} \) relates to \( x \), which is present in our original integrand.

Once we choose our substitution, we find its differential. So, if \( u = x^{2} \), differentiate both sides to get \( du = 2x \, dx \). Our integral involves \( x \, dx \), which we can relate to \( du \) by rearranging to get \( x \, dx = \frac{1}{2} du \).

• Pick a substitution: \( u = x^{2} \)
• Differential: \( du = 2x \, dx \)
• Rearrange: \( x \, dx = \frac{1}{2} du \)

Being able to express part of the original integrand in terms of \( du \) is key. It allows us to transform the integral into a simpler form: \( \int \sin(u) \cdot \frac{1}{2} du \).

Once we've successfully used substitution, we often encounter trigonometric integrals. These are integrals involving trigonometric functions like sine or cosine. In this example, after substitution, we face \( \int \sin(u) du \). Integrating trigonometric functions is straightforward when you know the basic rules.

The integral of \( \sin(u) \) is \( -\cos(u) \). Therefore, our integral simplifies to \( \frac{1}{2} \cdot (-\cos(u)) \) or \( -\frac{1}{2} \cos(u) \).

• Integral of \( \sin(u) \): \( -\cos(u) \)
• Apply the constant: \( \frac{1}{2} \)
• Result: \( -\frac{1}{2} \cos(u) \)

Understanding these basic integrals is critical. Knowledge of these foundational integrals allows us to tackle more complicated expressions with ease.

The constant of integration, often represented by \( C \), is a fundamental element of indefinite integrals. When we integrate a function, we are essentially finding a family of functions that all differ by a constant. That’s why after integration, we always add \( + C \).

In this exercise, after integrating \( \frac{1}{2} \int \sin(u) \) and simplifying to find \( -\frac{1}{2} \cos(u) \), we add \( C \). So our integral is \( -\frac{1}{2} \cos(u) + C \).

• Why do we add \( + C \)?
• Indefinite integrals represent a family of functions.
• \( C \) reflects a vertical shift in the family.

Always remember: the constant of integration is crucial when solving indefinite integrals. It ensures that we account for all possible solutions to the problem.