Problem 37 - Biocalculus Calculus for the Life Sciences

Question

\(37-38\) Find the general indefinite integral. Illustrate by graphing several members of the family on the same screen. \(\int\left(\cos x+\frac{1}{2} x\right) d x\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

The problem asks for the indefinite integral of the function \( \cos x + \frac{1}{2} x \). This function consists of two terms: the cosine function \( \cos x \) and the linear function \( \frac{1}{2} x \).

Step 2: Integrate Each Term Separately

Since the integral is a sum of two terms, integrate each term separately: - The integral of \( \cos x \) is \( \sin x \) because the derivative of \( \sin x \) is \( \cos x \).- The integral of \( \frac{1}{2} x \) is \( \frac{1}{4} x^2 \) because the derivative of \( \frac{1}{4} x^2 \) is \( \frac{1}{2} x \).

Step 3: Write the General Indefinite Integral

Combine the integrals from Step 2 to form the general solution with an integration constant \( C \):\[ \int (\cos x + \frac{1}{2} x)\, dx = \sin x + \frac{1}{4} x^2 + C \].

Step 4: Illustrate with Graphs

Illustrate the indefinite integral by plotting several curves corresponding to different values of \( C \) (say, \( C = -1, 0, 1 \)). Each plot will be a member of the family of curves given by:\[ y = \sin x + \frac{1}{4} x^2 + C \]. These graphs will shift vertically depending on the value of \( C \).

Key Concepts

Integration Techniques

When dealing with indefinite integrals, different integration techniques can simplify the process of solving them. Here, we are given the function \[ \int(\cos x + \frac{1}{2}x)\,dx \] This is a sum of two simpler functions: the trigonometric function \( \cos x \) and the linear function \( \frac{1}{2}x \). To integrate this, you would apply the rule that the integral of a sum is the sum of the integrals of each component.

For \( \cos x \), you recall the basic integration rule:

• The integral of \( \cos x \) is \( \sin x \), since the derivative of \( \sin x \) is \( \cos x \).

And for the linear term \( \frac{1}{2}x \) you use the power rule:

• The integral of \( \frac{1}{2}x \) is \( \frac{1}{4}x^2 \), because you add one to the exponent (making \( x^2 \)) and divide the coefficient by the new exponent.

After integrating each part separately, combine your results and don't forget to add the constant of integration \( C \), which represents a family of functions.

Graphical Representation

Graphical representation of indefinite integrals helps visualize the family of functions that represent the solution to the integral. By plotting the integrated function, you can understand how changes to the integration constant \( C \) impact the graph.

For our solution, \[ y = \sin x + \frac{1}{4}x^2 + C \] plotting this shows how each member of the family moves vertically on a graph depending on the value of \( C \). Start by plotting for several values like \( C = -1, 0, 1 \).

You'll notice:

• Each graph is a smooth curve.
• The sine component causes oscillation, while the quadratic term determines the overall shape.
• Different values of \( C \) cause the entire curve to shift up or down.

This visual approach gives you intuition about the effects of the constant \( C \) and how the graph approaches its behavior at different portions of the curve.

Family of Functions

In the context of indefinite integrals, understanding the term 'family of functions' is fundamental. This refers to the multiple solutions or functions obtained due to the constant of integration \( C \).

Whenever you solve an indefinite integral, the solution is not a single curve but a complete set of curves consisting of all possible values of \( C \). These curves share a similar shape but are essentially different from one another by a constant shift along the y-axis.

In our specific example, \[ y = \sin x + \frac{1}{4} x^2 + C \] represents a family when \( C \) differs.

• All functions in the family have the same curvature and oscillation pattern.
• The term \( C \) allows flexibility, making the graph adjustable to fit various applications and conditions.
• Illustrating with different constants helps understand how solutions can be manipulated by altering \( C \).

Understanding this helps highlight why the indefinite integral gives a broad form of solution, and not a precise value as found in definite integrals.