23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ ((1 + √x)/x)dx

Verified step by step guidance

1. Rewrite the integrand to simplify the expression. Split the fraction into two terms: \( \frac{1}{x} + \frac{\sqrt{x}}{x} \). Simplify further to get \( \frac{1}{x} + x^{-1/2} \).
2. Express the integrand in terms of powers of \(x\): \( x^{-1} + x^{-1/2} \). This makes it easier to apply the power rule for integration.
3. Apply the power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \), where \(n \neq -1\). For \(x^{-1}\), recall that \( \int x^{-1} dx = \ln|x| + C \).
4. Integrate each term separately: \( \int x^{-1} dx = \ln|x| \) and \( \int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2} \).
5. Combine the results to write the final integral: \( \ln|x| + 2x^{1/2} + C \), where \(C\) is the constant of integration. Check your work by differentiating this result to ensure it matches the original integrand.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation, where we seek a function F(x) such that F'(x) equals the integrand.

Integration Techniques

To solve integrals, various techniques can be employed, such as substitution, integration by parts, or partial fraction decomposition. In this case, simplifying the integrand, which is a rational function, can make the integration process more manageable. Recognizing patterns and applying appropriate techniques is crucial for finding the correct antiderivative.

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative obtained. This step ensures that the differentiation of the antiderivative returns the original integrand. This verification process is a fundamental practice in calculus, confirming the correctness of the integration performed.