Problem 24 - Thomas Calculus

Question

In Exercises \(17-56,\) find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(\frac{1}{5}-\frac{2}{x^{3}}+2 x\right) d x$$

Step-by-Step Solution

Answer

\( \int \left( \frac{1}{5} - \frac{2}{x^3} + 2x \right) \, dx = \frac{1}{5}x + \frac{1}{x^2} + x^2 + C \).

Step 1: Break Down the Integral

The given integral is \( \int \left( \frac{1}{5} - \frac{2}{x^3} + 2x \right) \, dx \). We'll separate the integral into three parts: \( \int \frac{1}{5} \, dx \), \( \int \left( -\frac{2}{x^3} \right) \, dx \), and \( \int 2x \, dx \). This makes the calculation easier.

Step 2: Solve the First Integral

The first integral is \( \int \frac{1}{5} \, dx \). The antiderivative of a constant \( c \) is \( cx \), hence the antiderivative is \( \frac{1}{5}x \).

Step 3: Solve the Second Integral

The expression \( -\frac{2}{x^3} \) is the same as \( -2x^{-3} \). The antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) where \( n eq -1 \). Therefore, the antiderivative is \( -2 \cdot \frac{x^{-2}}{-2} = x^{-2} = \frac{1}{x^2} \).

Step 4: Solve the Third Integral

The third integral is \( \int 2x \, dx \). Using the power rule, the antiderivative is \( 2 \cdot \frac{x^2}{2} = x^2 \).

Step 5: Combine All Parts

Adding the antiderivatives from each part, we get the complete antiderivative: \( \int \left( \frac{1}{5} - \frac{2}{x^3} + 2x \right) \, dx = \frac{1}{5}x + \frac{1}{x^2} + x^2 + C \), where \( C \) is the constant of integration.

Step 6: Verify by Differentiation

Differentiate \( \frac{1}{5}x + \frac{1}{x^2} + x^2 + C \) to verify: \( \frac{d}{dx}[\frac{1}{5}x + \frac{1}{x^2} + x^2 + C] = \frac{1}{5} - \frac{2}{x^3} + 2x \). This matches the original integrand, confirming our solution is correct.

Key Concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is essentially the reverse process of differentiation. When you find an antiderivative of a function, you are seeking another function whose derivative will give you the original function. This process is crucial because it helps in understanding the accumulation of quantities, such as areas under curves and displacement given velocity.

In the exercise you encountered, we were finding the antiderivative of the function:

• \( \frac{1}{5} - \frac{2}{x^3} + 2x \)

To find the antiderivative, we broke it into simpler parts, making it easier to manage each term individually. This piecemeal approach allows for systematic evaluation, ensuring that each segment of the expression is properly handled. The work is then unified by summing the individual antiderivatives.

Power Rule

The power rule is a fundamental technique used in calculus to find antiderivatives and derivatives. When dealing with powers of \(x\), we use it to simplify our calculations. For integration, the power rule states that the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1}\), provided \(n eq -1\).

In our solution, we use the power rule to evaluate terms such as \(2x\) and \(-\frac{2}{x^3}\). By transforming \(-\frac{2}{x^3}\) into \(-2x^{-3}\), we can apply:

• The power rule for integration gives us \(\frac{x^{n+1}}{n+1}\).
• For \(-2x^{-3}\), its antiderivative becomes \(-2 \cdot \frac{x^{-2}}{-2} = \frac{1}{x^2}\).

By the same rule, for \(2x\), the solution becomes \(\frac{x^2}{1}\). This application of the power rule is an essential skill in calculus, streamlining complex expressions.

Constant of Integration

Every time you compute an indefinite integral, you introduce a constant of integration, denoted as \(C\). Unlike definite integrals, which yield a specific number, indefinite integrals result in a general function. Without the constant of integration, the solution would refer only to one specific antiderivative, whereas there could be infinitely many shifts upward or downward on the graph.

In our solution, once we summed the antiderivatives of individual terms:

• \( \frac{1}{5}x + \frac{1}{x^2} + x^2 \)

we added \(+ C\) to include this constant of integration. This ensures that our solution encompasses all possible antiderivatives of the given function, offering the most general form of the solution.

Differentiation Verification

Differentiation verification is the process of differentiating an antiderivative to check if it matches the original function. This is a critical step because it confirms the correctness of your indefinite integral calculation. By taking the derivative of the function that we've found, we can ensure that it returns to the original integrand.

In verifying the solution \(\frac{1}{5}x + \frac{1}{x^2} + x^2 + C\), we perform the differentiation:

• The derivative of \( \frac{1}{5}x \) is \( \frac{1}{5} \).
• The derivative of \( \frac{1}{x^2} \) is \( -\frac{2}{x^3} \).
• The derivative of \( x^2 \) is \( 2x \).
• The derivative of \( C \), being a constant, is 0.

By adding these results, we reconstruct the original function \( \frac{1}{5} - \frac{2}{x^3} + 2x \), verifying our solution method and results.