Problem 16 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(12 x^{3}+3 x^{2}-5\right) d x\)

Step-by-Step Solution

Answer

The integral is \(3x^4 + x^3 - 5x + C\).

Step 1: Identify the Integral Components

We need to integrate the polynomial function \(12x^3 + 3x^2 - 5\). Each term of this polynomial will be integrated separately.

Step 2: Apply the Power Rule of Integration

The power rule for integration states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. Apply this rule to each term of the polynomial:\- For \(12x^3\), integrate to get \(12 \cdot \frac{x^{4}}{4} = 3x^4\).\- For \(3x^2\), integrate to get \(3 \cdot \frac{x^{3}}{3} = x^3\).\- For the constant \(-5\), since \(\int c \, dx = cx + C'\), this becomes \(-5x\).

Step 3: Combine the Integrated Terms and Add Constant of Integration

Combine all the integrated terms from Step 2: \\(3x^4 + x^3 - 5x\). Don't forget to add the constant of integration \(+ C\) which accounts for any constant that could have been differentiated to 0.\Thus, the integrated function is \(3x^4 + x^3 - 5x + C\).

Key Concepts

Power Rule of Integration

When we integrate polynomials, such as the given exercise, the power rule of integration is extremely useful. The power rule states that the integral of a term in the form of \(x^n\) is \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\). This handy formula applies to any power \(n\) that is not equal to -1.

To apply this rule effectively, follow these steps:

• Identify the power of \(x\) in your term, which is \(n\).
• Add one to this power to get \(n+1\).
• Divide the term by this new power \(n+1\).

For example, if you have a term like \(12x^3\), you add 1 to 3 to get 4, and then divide by 4. So, the integrated term becomes \(3x^4\). The power rule simplifies integrating polynomial terms immensely.

Polynomial Integration

Polynomial integration involves integrating each term of the polynomial separately. A polynomial might seem complex, but by breaking it down into individual terms and applying the power rule, it becomes quite manageable.

Consider the polynomial from the exercise: \(12x^3 + 3x^2 - 5\). Each term can be approached individually, integrating them one at a time:

• For \(12x^3\), use the power rule, integrating to \(3x^4\).
• For \(3x^2\), apply the power rule to get \(x^3\).
• The constant term \(-5\) is integrated to \(-5x\), because the integral of a constant \(c\) is \(cx + C'\).

Once each component has been integrated, simply combine the results to find the complete indefinite integral, making sure to include the constant of integration. This step-by-step approach simplifies polynomial integration significantly.

Constant of Integration

An essential part of indefinite integration is the constant of integration, often symbolized as \(C\). When finding indefinite integrals, it's important to remember that integration is the reverse process of differentiation.

Since a constant can disappear when differentiating (as its derivative is zero), the constant of integration compensates for any constant that might have originally been part of the function before differentiation.

For example, if \(f(x)\) is an antiderivative of \(F(x)\), then \(f(x) + C\) is an entire family of antiderivatives. Even though initially each term is integrated separately, adding \(C\) to the final result ensures the solution is complete. So for \(\int(12x^3 + 3x^2 - 5) \, dx\), the inclusion of \(+ C\) after combining the integrated terms \(3x^4 + x^3 - 5x\) is essential.
This inclusion ensures the accuracy and completeness of indefinite integrals.