Problem 31 - Applied Calculus

Question

Find each indefinite integral. \(\int \frac{6 x^{3}-6 x^{2}+x}{x} d x\)

Step-by-Step Solution

Answer

\(2x^3 - 3x^2 + x + C\)

Step 1: Simplify the Integrand

First, we divide each term in the numerator by the denominator. The original function is \(\frac{6x^3 - 6x^2 + x}{x}\). Simplify each term separately: \(\frac{6x^3}{x} = 6x^2\), \(\frac{-6x^2}{x} = -6x\), and \(\frac{x}{x} = 1\). The simplified integrand becomes: \(6x^2 - 6x + 1\).

Step 2: Integrate Each Term

We now find the indefinite integral of the simplified function \(\int (6x^2 - 6x + 1) \, dx\). Integrate each term separately: - The integral of \(6x^2\) is \(6 \cdot \frac{x^{3}}{3} = 2x^3\).- The integral of \(-6x\) is \(-6 \cdot \frac{x^2}{2} = -3x^2\).- The integral of \(1\) is \(x\).Putting it all together, we get \(2x^3 - 3x^2 + x\).

Step 3: Add the Constant of Integration

Since we are finding the indefinite integral, we need to include the constant of integration, \(C\). The final solution is: \(2x^3 - 3x^2 + x + C\).

Key Concepts

Simplifying the Integrand

Indefinite integrals can often appear overly complex at first glance, but there is a useful technique to make them more manageable: simplifying the integrand. By "integrand," we mean the function you are integrating. In this example, the original expression is \( \frac{6x^3 - 6x^2 + x}{x} \). Since this is a fraction, it suggests a division process that allows simplification.

To simplify, divide each term in the numerator by the denominator, \( x \). The process looks like this:

• \( \frac{6x^3}{x} = 6x^2 \)
• \( \frac{-6x^2}{x} = -6x \)
• \( \frac{x}{x} = 1 \)

Combining these results gives you the new, simpler integrand: \( 6x^2 - 6x + 1 \). This simplification makes the subsequent steps of integration much easier to handle.

Calculating the Integral

Once you've simplified the integrand, you're ready to calculate the indefinite integral. Integrating is essentially finding the antiderivative; this means finding a function whose derivative produces the original integrand. Our simplified integrand is \( 6x^2 - 6x + 1 \).

We integrate each term separately:

• The antiderivative of \( 6x^2 \) is calculated by adding 1 to the exponent (getting \( x^3 \)) and dividing by the new exponent: \( \frac{6}{3}x^3 = 2x^3 \).
• For \( -6x \), increase the exponent (turn \( x \) into \( x^2 \)) and divide: \( \frac{-6}{2}x^2 = -3x^2 \).
• The antiderivative of a constant like 1 is simply \( x \).

Combine these terms to get the integrated function: \( 2x^3 - 3x^2 + x \). This step reflects how smoothly integration can proceed when tackled methodically.

Adding the Constant of Integration

In an indefinite integral, you don't have bounds, which means the result is a family of functions. They differ by a constant value. This is captured by adding a \( C \) to the end result, known as the "constant of integration."

When you antiderivate a function, any constant you add will have a zero derivative. Thus, it "disappears" when differentiating, which means you need to introduce it back when integrating.

In our case, after integrating \( 6x^2 - 6x + 1 \), we got \( 2x^3 - 3x^2 + x \). The final step is to append \( C \) to this result, making the full solution \( 2x^3 - 3x^2 + x + C \). This constant is crucial since it makes the function general and reflects the full set of antiderivatives.