Problem 19 - Calculus: Early Transcendentals

Question

Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \( \displaystyle \int \biggl( \cos x + \frac{1}{2}x \biggr) \,dx \)

Step-by-Step Solution

Answer

The general indefinite integral is \( \int (\cos x + \frac{1}{2}x) \, dx = \sin x + \frac{x^2}{4} + C \).

Step 1: Identify each term in the integrand

The function you need to integrate is \( \cos x + \frac{1}{2}x \). This means the integrand consists of two terms: \( \cos x \) and \( \frac{1}{2}x \).

Step 2: Apply the sum rule for integration

The sum rule for integration allows us to integrate each term separately. Therefore, \( \int (\cos x + \frac{1}{2}x) \, dx = \int \cos x \, dx + \int \frac{1}{2}x \, dx \).

Step 3: Integrate each term separately

Integrate each term on its own:- The integral of \( \cos x \) is \( \sin x \).- For \( \int \frac{1}{2}x \, dx \), factor out the constant \( \frac{1}{2} \) to get \( \frac{1}{2} \int x \,dx \). Applying the power rule, the integral of \( x \) is \( \frac{x^2}{2} \), so \( \int \frac{1}{2}x \, dx = \frac{1}{2} \cdot \frac{x^2}{2} = \frac{x^2}{4} \).

Step 4: Combine the results of the integration

Add the results of the integration from the previous step: \[ \int \left(\cos x + \frac{1}{2}x\right) \, dx = \sin x + \frac{x^2}{4} + C \] where \( C \) is the constant of integration.

Step 5: Sketch several members of the family of curves

To graph several members of the family, consider different values of the constant \( C \). For example, you can plot \( \sin x + \frac{x^2}{4} + C \) for \( C = -2, 0, 2, 4 \). Each value of \( C \) will shift the graph vertically, creating a variety of curves on your plot.

Key Concepts

Sum Rule for Integration

When dealing with the integral of a sum of functions, the process can be made much easier by using the sum rule for integration. This rule states that the integral of the sum of two or more functions is the same as the sum of their individual integrals. In other words, if you have a function expressed as

• \( f(x) + g(x) \)

then

• \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \)

This allows us to break down complex problems into simpler parts. By integrating each term separately, we can focus on one function at a time.
In the example given, the function \( \cos x + \frac{1}{2}x \) can be split into the two separate terms \( \cos x \) and \( \frac{1}{2}x \).
This makes the integration much more manageable.

Constant of Integration

In indefinite integrals, there isn’t just one solution, but rather a family of solutions. This is captured by the constant of integration, denoted as \( C \). Whenever you find an indefinite integral, you must add this constant. Why? Because when you differentiate a constant, the result is zero. Hence, any constant could have been in the original function before differentiation.
When you see an indefinite integral result like

• \( \sin x + \frac{x^2}{4} + C \),

\( C \) represents an infinite number of constant values. Each choice of \( C \) creates a different curve that vertically shifts on the graph, reflecting all possible scenarios of the antiderivative.
This concept is key when not all variables are known about a function.

Power Rule

One of the most frequently used rules in integration and differentiation is the power rule. The power rule for integration is straightforward and is particularly useful when integrating polynomials. The general form of this rule is:

• If \( n eq -1 \), then \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

Applying the power rule transforms powers of \( x \) into the antiderivative, by increasing the exponent by one and dividing by this new exponent.
In practice, let’s see this rule applied to \( \int x \, dx \):

• Here, the exponent \( n = 1 \). So, you compute \( \frac{x^{1+1}}{1+1} = \frac{x^2}{2} \).

This simple rule is powerful for simplifying integrals quickly and is essential for handling expressions with polynomial terms.