Problem 39 - Calculus and its applications

Question

Find each integral. $$ \int\left(x^{2}-\frac{3}{2} \sqrt{x}+x^{-4 / 3}\right) d x $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Integral

First, split the given integral into three separate integrals for each term in the expression: \( \int x^2 \, dx \), \( \int -\frac{3}{2} \sqrt{x} \, dx \), and \( \int x^{-4/3} \, dx \). This will allow us to integrate each term individually.

Step 2: Integrate the First Term

Use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for any real number \( n eq -1 \). Apply it to \( \int x^2 \, dx \): \[ \int x^2 \, dx = \frac{x^{3}}{3}. \]

Step 3: Integrate the Second Term

For \( \int -\frac{3}{2} \sqrt{x} \, dx \), recognize \( \sqrt{x} = x^{1/2} \). Use the power rule again: \( \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} \). Thus, \[ \int -\frac{3}{2} x^{1/2} \, dx = -\frac{3}{2} \cdot \frac{2}{3} x^{3/2} = -x^{3/2}. \]

Step 4: Integrate the Third Term

For \( \int x^{-4/3} \, dx \), apply the power rule: \( \int x^{-4/3} \, dx = \frac{x^{-4/3 + 1}}{-4/3 + 1} \). Simplifying gives \[-\frac{x^{-1/3}}{-1/3} = -3x^{-1/3}. \]

Step 5: Combine the Results

Combine the results of the three integrals to form the final solution: \[ \int \left(x^{2}-\frac{3}{2} \sqrt{x}+x^{-4 / 3}\right) dx = \frac{x^3}{3} - x^{3/2} - 3x^{-1/3} + C, \] where \( C \) is the constant of integration.

Key Concepts

Power rule for integration

The power rule for integration is a fundamental tool used to solve integrals involving polynomial expressions. This rule applies when integrating terms of the form \( x^n \), where \( n \) is any real number except \( -1 \). For such an expression, the rule is expressed as:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Here, \( C \) represents the constant of integration, signifying that indefinite integrals can have an infinite number of solutions that differ by a constant value. This rule simplifies what might otherwise require cumbersome calculations, especially as it neatly accommodates most polynomial terms.
If applied correctly, it makes integrating powers of \( x \) straightforward:

• Add 1 to the power \( n \).
• Divide the term by the new power.

The power rule streamlines much of calculus by providing a direct way to compute antiderivatives for many real-world functions encountered in algebra and beyond.

Definite and indefinite integrals

Understanding integrals involves differentiating between two main types: definite and indefinite integrals. Indefinite integrals, like those examined in typical calculus exercises, represent a family of functions and include a constant of integration \( C \). For example, integrating \( x^2 \) gives the general form \( \int x^2 \, dx = \frac{x^3}{3} + C \). These provide a solution that includes all possible antiderivatives of a function.
On the other hand, definite integrals calculate the net area under a curve between two specific points, typically yielding a numeric value. For instance, \( \int_{a}^{b} x^2 \, dx = F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( x^2 \). This application is pivotal in physics and engineering when determining displacement, area, and total accumulation.
While indefinite integrals focus on finding a general solution without specific boundary conditions, definite integrals need exact values and often tell us much about the real-world contexts of the functions being analyzed.

Polynomial integration

Polynomial integration involves taking the integral of polynomial expressions, which are combinations of terms with non-negative integer exponents on \( x \). In a polynomial, each term can be integrated separately by applying the power rule.
Consider a polynomial like \( ax^n + bx^m + c \). To integrate, process each term individually:

• Apply the power rule to each term: For each \( ax^n \), the integral becomes \( \frac{a}{n+1}x^{n+1} \).
• If a term is constant, like \( c \), its integral is simply \( cx \).

The result for the integral of a polynomial function is a new polynomial function that represents the family of antiderivatives. In our original exercise, breaking down each term in \( x^2 - \frac{3}{2}\sqrt{x} + x^{-4/3} \) and integrating them using the power rule led to the antiderivative \( \frac{x^3}{3} - x^{3/2} - 3x^{-1/3} + C \).
Thus, polynomial integration makes extensive use of simple principles applied repetitively, ensuring that we can decipher more complex relationships within polynomial expressions.