Problem 36 - Biocalculus Calculus for the Life Sciences

Question

Verify by differentiation that the formula is correct. \(\int x \cos x d x=x \sin x+\cos x+C\)

Step-by-Step Solution

Answer

The derivative of the solution matches the integrand, thus verifying the formula is correct.

Step 1: Understand the Given Function

You're given the integral \( \int x \cos x \, dx \) with a purported solution: \( x \sin x + \cos x + C \). To verify it, we must differentiate the proposed solution.

Step 2: Differentiate the Proposed Solution

Differentiate the function \( f(x) = x \sin x + \cos x + C \). Using the product rule for \( x \sin x \) and the derivative of \( \cos x \), compute:1. Derivative of \( x \sin x \): \( \frac{d}{dx}[x \sin x] = \sin x + x \cos x \) (using product rule)2. Derivative of \( \cos x \): \( \frac{d}{dx}[\cos x] = -\sin x \)3. Derivative of \( C \): \( \frac{d}{dx}[C] = 0 \) since \( C \) is a constant.

Step 3: Simplify the Derivative

Combine the derivatives found in Step 2:\( \frac{d}{dx}[x \sin x + \cos x + C] = \sin x + x \cos x - \sin x = x \cos x \).

Step 4: Match the Result with the Original Integrand

The simplified derivative \( x \cos x \) matches the original integrand of the integral \( \int x \cos x \, dx \). Since the derivative of the solution equals the original function, the formula is verified.

Key Concepts

Differentiation

Differentiation is a cornerstone of calculus, allowing us to find the rate at which a function is changing at any given point. In simple terms, differentiation transforms a function into its derivative. This plays a key role in many real-world applications, such as finding velocities from position-time graphs or slopes of curves.

To differentiate a function, you typically apply certain rules based on the function's form, like power rules, chain rules, or in this case, the product rule. The derivative tells us how much the function value (or output) changes as the input changes, also often referred to as the slope of the function at a point.

When verifying solutions, like in our exercise, differentiation is used to check if the proposed function, when differentiated, gives back the original function we started with before integration. This confirms that the integration was done correctly.

Product Rule

The product rule is a specific method in differentiation used when a function is the product of two separate functions. It helps in finding the derivative when two functions are multiplied together. The formula for the product rule states: if you have two functions, say, \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

• \( \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

In the exercise, this rule applies to \( x \sin x \). Here:

• \( u(x) = x \) and \( v(x) = \sin x \)
• Differentiate \( u(x) \) to get \( u'(x) = 1 \)
• Differentiate \( v(x) \) to get \( v'(x) = \cos x \)

Using the product rule, we find the derivative to be \( \sin x + x \cos x \).
This method is essential because it correctly accounts for the change in both parts of the product, giving a complete picture of how the overall function behaves.

Definite Integrals

Definite integrals are integral calculus tools that compute the signed area under a curve, bounded by specific limits on the x-axis. While indefinite integrals result in a general form expression with an arbitrary constant \( C \), definite integrals result in an actual numerical value.

The primary purpose of definite integration is to determine accumulated quantities, such as total displacement from a velocity function or total revenue from a sales rate function over a given period.

In our exercise, we are dealing with verification through indefinite integration, as the original function does not have specified limits. However, understanding definite integrals is crucial because it sets a boundary for these calculations, turning them from generalized solutions into specific, applicable answers to problems.