Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int\left(\frac{1}{x^{2}}-\frac{1}{\sqrt[3]{x^{2}}}+\frac{1}{\sqrt{x}}\right) d x$$

Step-by-Step Solution

Answer

\(-\frac{1}{x} + 3x^{\frac{1}{3}} + 2x^{\frac{1}{2}} + C\)

Step 1: Rewrite the given integral

First, rewrite the given integral in terms of the power of x: \[ \int\left(x^{-2}-x^{-\frac{2}{3}}+x^{-\frac{1}{2}}\right)dx \] Now, we will find the integral of each term and add them up.

Step 2: Integrate the first term

To find the integral of the first term, use the power rule for integration: \[ \int x^{-2}dx = \frac{x^{-1}}{-1} + C_1 = -\frac{1}{x}+C_1 \]

Step 3: Integrate the second term

Integrate the second term using the power rule for integration: \[ \int x^{-\frac{2}{3}}dx = \frac{x^{\frac{1}{3}}}{\frac{1}{3}} + C_2 = 3x^{\frac{1}{3}} + C_2 \]

Step 4: Integrate the third term

Integrate the third term, again using the power rule for integration: \[ \int x^{-\frac{1}{2}}dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C_3 = 2x^{\frac{1}{2}} + C_3 \]

Step 5: Add the integrated terms and combine constants

Add up the integrated terms from Steps 2, 3, and 4 and combine the constants: \[ -\frac{1}{x} + 3x^{\frac{1}{3}} + 2x^{\frac{1}{2}} + (C_1 + C_2 + C_3) \] Since \(C_1\), \(C_2\), and \(C_3\) are all constants, we can rewrite their sum as a single constant, say C: \[ -\frac{1}{x} + 3x^{\frac{1}{3}} + 2x^{\frac{1}{2}} + C \] This is the indefinite integral of the given function.

Key Concepts

Introduction to Integration Techniques

Integration is a fundamental concept in calculus, important for understanding the accumulation of quantities and the area under curves. Integration techniques are mathematical methods used to find antiderivatives or indefinite integrals—functions which generate the original mathematical expressions when differentiated.

There are various techniques for integrating different kinds of functions, including but not limited to, the power rule, integration by parts, substitution, and partial fractions. Mastering these techniques is crucial for solving complex integrals that you may encounter in calculus. Each technique is suitable for a specific type of function, and sometimes a combination of these methods must be used to find a solution.

Power Rule for Integration

A common tool in the calculus toolkit is the power rule for integration. This rule applies to functions of the form \( x^n \), where \( n \) is any real number except -1.

The rule states that: \[\int x^n dx = \frac{x^{n+1}}{n+1} + C\]where \( C \) represents the constant of integration. This method is derived from the reverse process of differentiation. Remember that the power rule for integration doesn't work when \( n = -1 \), in which case the integral is a natural logarithm. This rule simplifies the integration process immensely, making it possible to integrate polynomial expressions term by term.

Understanding Antiderivatives

In calculus, antiderivatives are functions that