Problem 67 - Applied Calculus

Question

Find the indefinite integrals. $$\int\left(x^{5}-12 x^{3}\right) d x$$

Step-by-Step Solution

Answer

\( \frac{x^6}{6} - 3x^4 + C \)

Step 1: Identify the Integral

The given integral is \( \int (x^5 - 12x^3) \, dx \). This is an indefinite integral which means we need to find the antiderivative.

Step 2: Apply the Power Rule

To integrate each 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 \( n eq -1 \).

Step 3: Integrate the First Term

The first term is \( x^5 \). Applying the power rule, integrate it to get \( \frac{x^{5+1}}{5+1} = \frac{x^6}{6} \).

Step 4: Integrate the Second Term

The second term is \(-12x^3\). Apply the power rule to \(-12x^3\) to get \(-12 \cdot \frac{x^{3+1}}{3+1} = -12 \cdot \frac{x^4}{4} = -3x^4 \).

Step 5: Write the General Solution

Combine the results from Step 3 and Step 4 to form the general solution of the integral. Thus, \( \int (x^5 - 12x^3) \, dx = \frac{x^6}{6} - 3x^4 + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

When integrating polynomials, such as the terms present in the expression \( x^5 - 12x^3 \), a useful tool is the power rule for integration. This rule simplifies the process of finding antiderivatives by providing a straightforward formula:

• For any term \( x^n \), where \( n \) is a real number except for \( -1 \), the antiderivative is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

In our example, we apply this formula to each term separately. First, integrate the term \( x^5 \) by adding 1 to the exponent, leading to \( \frac{x^{6}}{6} \). Similarly, for the term \(-12x^3\), applying the same rule results in \(-3x^4\). This step-by-step approach is crucial in simplifying polynomials into their integral form.

Antiderivative

The antiderivative, also known as the indefinite integral, reverses the process of differentiation. Essentially, it answers the question: What function differentiates to give the original function or expression?

In the context of finding \( \int (x^5 - 12x^3) \, dx \), we seek a function whose derivative is \( x^5 - 12x^3 \). By applying the power rule for each term, we discovered that

• The antiderivative of \( x^5 \) is \( \frac{x^6}{6} \).
• The antiderivative of \(-12x^3\) results in \(-3x^4\).

Combining these results gives us the general solution \( \frac{x^6}{6} - 3x^4 + C \). This solution isn't just one function; it represents a family of functions, each differing by a constant \( C \).

Constant of Integration

In indefinite integrals, we often encounter the term 'constant of integration,' represented by \( C \).

The constant of integration reflects the fact that antiderivatives are not unique. Consider that differentiating a constant results in zero, making it impossible to determine an original constant if it existed. Thus, when finding indefinite integrals, we include \( C \) to represent any potential constant that was "lost" during differentiation.

• For example, when integrating \( x^5 - 12x^3 \), the solution is expressed as \( \frac{x^6}{6} - 3x^4 + C \).

The constant \( C \) means that every function within the family of functions represented by the solution can differ by a constant value. This is crucial in applications that require specific initial conditions to find a unique solution to a problem. By including \( C \), we ensure that we accommodate all possible original functions.