Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x$$

Step-by-Step Solution

Answer

The indefinite integral of the given function is: \( \int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x = \frac{3}{2}x^4 - \frac{3}{x} - \frac{1}{2}x^2 + C \)

Step 1: Break the function into individual terms

We are given the function to integrate: \( \int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x \) To find the integral, we need to break this function into individual terms: \( \int{6x^3dx} + \int{\frac{3}{x^2}dx} - \int{xdx} \)

Step 2: Integrate each term

Now, we integrate each term separately: For the first term: \( \int{6x^3dx} \) Use the power rule: \(\int{x^n dx} = \frac{x^{n+1}}{n+1} + C\) So, \( \int{6x^3dx} = 6\frac{x^{3+1}}{3+1} + C_1 = \frac{3}{2}x^4 + C_1 \) For the second term: \( \int{\frac{3}{x^2}dx} \) Rewrite the function as: \( \int{3x^{-2}dx} \) Use the power rule again: \( \int{3x^{-2}dx} = 3\frac{x^{-2+1}}{-2+1} + C_2 = -3x^{-1} + C_2 = -\frac{3}{x} + C_2 \) For the third term: \( \int{xdx} \) Using the power rule: \( \int{xdx} = \frac{x^{1+1}}{1+1} + C_3 = \frac{1}{2}x^2 + C_3 \)

Step 3: Combine the integrated terms

Now that we have integrated each term, we need to combine them to find the indefinite integral of the given function: \( \int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x = \frac{3}{2}x^4 - \frac{3}{x} - \frac{1}{2}x^2 + C \) where C is the constant of integration, equal to the sum of constants, i.e., C = C_1 + C_2 + C_3. So the indefinite integral of the given function is: \( \int\left(6 x^{3}+\frac{3}{x^{2}}-x\right) d x = \frac{3}{2}x^4 - \frac{3}{x} - \frac{1}{2}x^2 + C \)

Key Concepts

Indefinite Integral

Integrals are fundamental in calculus and help us find the area under curves and accumulated quantities. An *indefinite integral* is essentially the opposite of differentiation. It gives us a function whose derivative is the original function we started with. This function is called the *antiderivative*. An indefinite integral includes a constant of integration, often denoted as 'C', because the process of integration only finds one antiderivative — but there are infinitely many since constants vanish when differentiated.

When you see a problem like \( \int (6x^3 + \frac{3}{x^2} - x) dx \), the task is to determine the antiderivative for each term within the integral. The integral's symbol is an elongated 'S', standing for 'summation': it's used to denote the process of integration, where we are summing up an infinite number of infinitely small pieces.

Power Rule in Integration

The *Power Rule in Integration* is analogous to the Power Rule in Derivatives and is a very handy tool for finding antiderivatives of polynomial functions. The rule is straightforward: for any power of \( x \), denoted as \( x^n \), the integral is given by:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
This formula works as long as \( n eq -1 \), because when \( n = -1 \), we have a special case where the integral of \( x^{-1} \) is \( \ln|x| \). The Power Rule allows us to efficiently integrate each term of a polynomial separately. For example, to integrate \( 6x^3 \), apply the Power Rule:

• Increase the exponent by 1, turning \( x^3 \) into \( x^4 \).
• Divide by the new exponent, resulting in \( \frac{6}{4}x^4 \) which simplifies to \( \frac{3}{2}x^4 \).

Remember, each term in a function can be integrated separately using this rule, which simplifies calculations greatly.

Integration of Polynomial Functions

*Polynomial functions* consist of multiple terms where each term is a constant multiplied by a variable raised to a non-negative integer power. Integrating polynomial functions involves applying the Power Rule to each individual term. For example, \( 6x^3 \) and \( -x \) in the function \( 6x^3 + \frac{3}{x^2} - x \) are already written with their powers, but \( \frac{3}{x^2} \) needs to be rewritten as \( 3x^{-2} \) for easier integration.

Thus after rewriting, you integrate each term separately:

• For \( 6x^3 \), you get \( \frac{3}{2}x^4 \).
• For \( 3x^{-2} \), the integral is \( -3x^{-1} \) or equivalently \( -\frac{3}{x} \).
• The term \( -x \) integrates to \( -\frac{1}{2}x^2 \).

Each of these calculations follows directly from applying the Power Rule.
Total integration becomes combining these integrated terms together, plus the constant of integration \( C \). The presence of \( C \) accounts for any constant that could be added to each antiderivative, signifying the continuous family of possible solutions.