Problem 19 - Essential Calculus

Question

Find the indefinite integral and check the result by differentiation. $$ \int d x $$

Step-by-Step Solution

Answer

The indefinite integral of \( \frac{1}{\sqrt{2x}} \) is \( 2 \sqrt{x} + C \)

Step 1: Re-write Function

We can rewrite the integrand \( \frac{1}{\sqrt{2x}} \) as \( \frac{1}{\sqrt{2} \sqrt{x}} \) or \( \frac{1}{\sqrt{2}} x^{-1/2} \) which now appears as a power function.

Step 2: Apply the Power Rule for integration.

Using the Power Rule for integration, our problem simplifies to \( \int \frac{1}{\sqrt{2}} x^{-1/2} dx \) which integrates to \( \frac{1}{\sqrt{2}} \frac{x^{1/2}}{1/2} + C \). Consequently, our solution simplifies to \( 2\sqrt{x} + C \).

Step 3: Differentiate to check answer.

Finally, to confirm the solution, differentiate \( 2\sqrt{x} + C \). The derivative of \( 2\sqrt{x} \) is \( \frac{1}{\sqrt{2x}} \), which is the original function, reaffirming the accuracy of our solution.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding a function when its derivative is given. This process can also be thought of as calculating the area under the curve of a given function on a graph.

The integral we are concerned with is an indefinite integral, which represents a family of functions that all differ by a constant. This is written with an integral sign and a function, followed by 'dx' which signifies the variable with respect to which we are integrating. For example, finding the indefinite integral of \(\frac{1}{\sqrt{2x}}\) involves identifying a function that, when differentiated, gives the original function \(\frac{1}{\sqrt{2x}}\). This is like reverse-engineering the process of differentiation.

Indefinite integrals are incredibly useful in various fields such as physics for finding quantities like displacement when velocity is known, or in economics for calculating total cost when marginal cost is given. Remember that when you solve an indefinite integral, you always add a constant of integration \(C\) since there are infinitely many antiderivatives that differ by a constant.

Power Rule for Integration

The power rule for integration is a technique used to integrate functions of the form \(x^n\), where \(n\) is any real number except -1. This rule states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.

Applying the Power Rule

In the given exercise, the integrand \(\frac{1}{\sqrt{2x}}\) was first rewritten as \(\frac{1}{\sqrt{2}}x^{-1/2}\) to fit the power function form. With this adjustment, applying the power rule becomes straightforward: increase the exponent by 1 to get \(-1/2 + 1 = 1/2\) and then divide by the new exponent. After simplifying, we get the antiderivative \(2\sqrt{x} + C\).

Students should note that the power rule simplifies the process of finding antiderivatives and is a fundamental tool for solving indefinite integrals involving power functions -- making the otherwise complex process a matter of applying a quick formula.

Differentiation

Differentiation, another core operation in calculus, is the process of finding the derivative of a function; it measures how a function changes as its input changes. The derivative represents the rate of change or the slope of the function at any given point.

Checking Work Through Differentiation

In the context of the exercise, differentiation serves as a means to check the correctness of the indefinite integral. After integrating \(\frac{1}{\sqrt{2x}}\) to find the function \(2\sqrt{x} + C\), we differentiate this result to see if it matches the original function. The derivative of \(2\sqrt{x}\) is \(\frac{1}{\sqrt{x}}\), which when simplified, gives us back our initial function. This step confirms the integration was performed correctly.

Understanding differentiation is crucial as it complements integration. Mastery of both these operations is essential for a thorough grasp of calculus and is instrumental in solving a wide array of problems in mathematics, physics, engineering, and beyond.