Problem 23 - Applied Calculus

Question

Find an antiderivative. $$k(x)=10+8 x^{3}$$

Step-by-Step Solution

Answer

The antiderivative is \( K(x) = 10x + 2x^4 + C \).

Step 1: Understand the Exercise

The task asks us to find an antiderivative (also known as an indefinite integral) for the function \( k(x) = 10 + 8x^3 \). This means we need to find a function whose derivative is \( 10 + 8x^3 \).

Step 2: Antiderivative of Constant Term

The first term in the function \( k(x) \) is \( 10 \). The antiderivative of a constant \( c \) is \( cx \). Therefore, the antiderivative of \( 10 \) is \( 10x \).

Step 3: Antiderivative of Power Term

The second term in \( k(x) \) is \( 8x^3 \). We use the power rule of integration, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Applying this to \( 8x^3 \), its antiderivative is \( \frac{8x^{4}}{4} = 2x^{4} \).

Step 4: Combine the Antiderivatives

To find the antiderivative of the entire function \( k(x) \), combine the results from the previous steps. That gives us \( K(x) = 10x + 2x^4 + C \), where \( C \) is the constant of integration.

Key Concepts

Indefinite Integral

When you see a problem asking for an antiderivative, it's referring to the process of finding an indefinite integral. This process involves determining a function that will result in the original function when differentiated. In the context of

• Fundamentally reversing differentiation.
• Providing a family of functions instead of a specific one, which is why we include a constant 'C'.

The indefinite integral doesn’t rely on fixed limits or boundaries. Instead, it embodies an entire class of functions that nicely encase the derivative function, lessening the focus on explicit numerical answers. In essence, it's about finding a broader solution for the original function. Remember, the signature mark of an indefinite integral is the addition of the constant of integration, ‘C,’ which expresses this generality.

Power Rule

The power rule is a foundation in both differentiation and integration processes. For differentiation, it provides a straightforward approach to deriving the power of any function. Integrating functions harnesses a broadened version of this principle. The power rule for integration operates via the formula:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).
• Simply increase the power by one, divide by this new power, and append ‘C’.

This rule is incredibly useful when dealing with polynomial expressions, where each term's power can be easily adjusted, allowing for straightforward computation of antiderivatives. In practice, whether integrating polynomials or other functions, it frees up the task of diving into complex calculations by offering a foolproof methodology.

Constant of Integration

Every time you calculate an indefinite integral, you'll need to add a constant of integration, denoted usually by 'C'. This may seem straightforward, but its purpose is deeply ingrained in calculus. When differentiating, tiny pieces of constant information are lost. The 'C' helps unearth these missing pieces when you switch back.

• Indispensable to accurately represent the entire solution set of functions.
• Predominantly characterized in equations as \( K(x) = 10x + 2x^{4} + C \).
• Ensures that any function derived from the integral retains its unique characteristics not captured just by coefficients.

In real-world problems, knowing this constant also allows for precise alignment of formulas with boundary conditions or historical values, ensuring calculations are comprehensive and informed.