Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{1+\sqrt{x}}{x} d x$$

Short answer

Question: Find the indefinite integral of the function $f(x) = \frac{1+\sqrt{x}}{x}$ Answer: The indefinite integral of the function is $\int \frac{1+\sqrt{x}}{x} dx = \ln|x| + 2\sqrt{x} + C$.

Step-by-step solution

Rewrite the function

We can rewrite the given function as the sum of two separate fractions: $$\int \frac{1+\sqrt{x}}{x} dx = \int \left(\frac{1}{x} + \frac{\sqrt{x}}{x}\right) dx$$ Now, simplify the second term: $$= \int \left(\frac{1}{x} + \frac{x^{\frac{1}{2}}}{x}\right) dx$$ $$= \int \left(\frac{1}{x} + x^{-\frac{1}{2}}\right) dx$$

Integrate each term

Now, we can integrate each term separately and apply the basic integration rules: $$= \int \frac{1}{x} dx + \int x^{-\frac{1}{2}} dx$$

Apply the Power Rule for Integration

Using the power rule for integration, which states that if \(n \neq -1\), then \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\). We will apply this rule to our integral: For the first term, \(n=-1\), so we use the special case rule: \(\int x^{-1} dx = \ln|x| + C\) For the second term, \(n=-\frac{1}{2}\): $$\int x^{-\frac{1}{2}} dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = 2\sqrt{x} + C$$

Combine the integrated terms

Now, combine the integrated results: $$\int \frac{1+\sqrt{x}}{x} dx = \ln|x| + 2\sqrt{x} + C$$

Check the result using differentiation

To confirm the correctness of our result, we will differentiate the integrated function: $$\frac{d}{dx}\left(\ln|x| + 2\sqrt{x} + C\right) = \frac{d}{dx}(\ln|x|) + \frac{d}{dx}(2\sqrt{x})+\frac{d}{dx}(C)$$ Using differentiation rules, we have: $$\frac{1}{x} + \frac{1}{\sqrt{x}} = \frac{1}{x}+\frac{\sqrt{x}}{x}$$ This matches the original function we wanted to integrate. Therefore, our integration result is correct: $$\int \frac{1+\sqrt{x}}{x} dx = \ln|x| + 2\sqrt{x} + C$$

Key concepts

Indefinite Integrals

Indefinite integrals are a central concept in calculus, representing the process of finding the most general form of an antiderivative for a given function. This means that, unlike definite integrals, indefinite integrals include a constant of integration, denoted by 'C'. This constant arises because differentiation of a constant yields zero. Hence, when finding the antiderivative of a function, it is essential to add this constant to account for any original constant term that was differentiated away.
For instance, when you integrate a function, the result is a family of functions unless more conditions (like initial values) are provided to specify a particular solution. Understanding indefinite integrals is key to solving problems involving area under curves or whenever a rate of change is involved.

Integration Techniques

Integration techniques refer to various methods used to solve integrals that cannot be readily identified from basic rules. These techniques increase the ability to integrate more complex functions.
Common integration techniques include:

• Substitution: This involves changing variables to simplify the integral.
• Integration by Parts: Via a rule derived from the product rule for differentiation, it helps in integrating products of functions.
• Partial Fraction Decomposition: Useful when integrating rational functions.

In the exercise provided, the technique involved rewriting the integral to separate the terms into simpler forms that can each be integrated individually with the basic integration rules, showcasing a strategic approach to handle integrals with challenging components.

Differentiation

Differentiation is the process of finding the derivative of a function. This derivative provides the rate at which the function changes at any point. It is fundamentally the inverse process of integration.
Using differentiation can confirm the correctness of an integration solution: if you differentiate the integrated function and get back the original function, then the integration was performed correctly. This is a valuable tool for verifying solutions. In the provided exercise, differentiation was used to verify that the integrated function, when differentiated, returned the integrand of the original integral, confirming the solution's accuracy.

Power Rule for Integration

The Power Rule for Integration is a fundamental rule for finding antiderivatives of power functions. It is essential for integrating expressions where the variables are raised to a power. The rule states:

• If the function has the form \( x^n \), then its integral is \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) assuming \( n eq -1 \).

This rule simplifies the process of finding antiderivatives for polynomials and power functions.
However, when \( n = -1 \), the rule changes slightly because the integral of \( x^{-1} \) is \( \ln|x| \). In the exercise, both these concepts were applied: the natural logarithm for \( 1/x \) and the power rule for \( x^{-1/2} \). These foundational rules make solving indefinite integrals much more manageable.