91Ó°ÊÓ

In calculus, the product rule is a fundamental technique used to find the derivative of a product of two functions. When you have two functions, say \( u(x) \) and \( v(x) \), and you need to find the derivative of their product, \( u(x) v(x) \), you can't just take the derivatives separately and multiply them. Instead, you use the product rule, which states:

• \( (uv)' = u'v + uv' \)

This formula helps in breaking down the problem by expressing it as the sum of two parts:

• First, differentiate \( u \) and multiply it by \( v \).
• Second, multiply \( u \) by the derivative of \( v \).

In the problem \( \sin x - x \cos x \), the product rule is applied to the term \( -x \cos x \):

• Here, \( u = x \) and \( v = \cos x \).
• The derivatives are \( u' = 1 \) and \( v' = -\sin x \).

Putting these into the product rule: \( (1)(\cos x) + (x)(-\sin x) = \cos x - x \sin x \). This simplification leads to finding the derivative accurately without errors.

Integration by parts is a powerful integration technique that's the counterpart to the product rule in differentiation. It is used when you're dealing with the integral of a product of two functions, commonly expressed in the form:

• \( \int u \, dv = uv - \int v \, du \)

The choice of \( u \) and \( dv \) is crucial for simplifying the integral. In the exercise, you're asked to integrate the expression \( \int x \sin x \, dx \). To apply integration by parts:

• Choose \( u = x \), hence \( du = dx \).
• Select \( dv = \sin x \, dx \), which means \( v = -\cos x \).

Then, using the integration by parts formula, you get:

• \( \int x \sin x \, dx = -x \cos x + \int \cos x \, dx \)

The remaining integral \( \int \cos x \, dx \) is straightforward, resulting in \( \sin x \). Thus, the complete integration is:

• \( \int x \sin x \, dx = -x \cos x + \sin x + C \)

Here, \( C \) is the constant of integration, important to remember in indefinite integrals.

Trigonometric functions such as \( \sin x \) and \( \cos x \) frequently appear in calculus problems. Understanding their properties and derivatives is key to solving a variety of problems.Let's quickly review:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).
• The integral of \( \cos x \) is \( \sin x + C \).
• The integral of \( \sin x \) is \( -\cos x + C \).

These rules are essentials when differentiating or integrating trigonometric functions. In the original exercise, differentiating \( \sin x - x \cos x \) involves both \( \sin x \) and \( \cos x \), making it necessary to know these derivatives by heart.Understanding these basic trigonometric derivatives and integrals will greatly aid in tackling more complex calculus problems, allowing you to convert a seemingly complicated problem into simple, manageable steps.