Problem 29 - 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. $$\int\left(8 y-\frac{2}{y^{1 / 4}}\right) d y$$

Step-by-Step Solution

Answer

The antiderivative is \( 4y^2 - \frac{8}{3}y^{3/4} + C \).

Step 1: Break Down the Integral

The given integral is \( \int \left(8y - \frac{2}{y^{1/4}}\right) \, dy \). We can split this into two separate integrals: \( \int 8y \, dy - \int \frac{2}{y^{1/4}} \, dy \).

Step 2: Integrate the First Term

For the integral \( \int 8y \, dy \), apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here, \( n = 1 \), so \( \int 8y \, dy = 8 \cdot \frac{y^{2}}{2} = 4y^{2} \).

Step 3: Rewrite the Second Term

For the integral \( \int \frac{2}{y^{1/4}} \, dy \), rewrite \( \frac{2}{y^{1/4}} \) as \( 2y^{-1/4} \). Now, you will integrate \( 2y^{-1/4} \).

Step 4: Integrate the Second Term

Apply the power rule for integration to \( \int 2y^{-1/4} \, dy \): \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). Here, \( n = -1/4 \), so \( \int 2y^{-1/4} \, dy = 2 \cdot \frac{y^{3/4}}{3/4} = \frac{8}{3}y^{3/4} \).

Step 5: Combine the Results

Combine the results from Steps 2 and 4: \( 4y^{2} - \frac{8}{3}y^{3/4} + C \). This is the most general form of the antiderivative.

Step 6: Check by Differentiation

Differentiate \( 4y^{2} - \frac{8}{3}y^{3/4} + C \). The derivative of \( 4y^2 \) is \( 8y \), and the derivative of \( - \frac{8}{3}y^{3/4} \) is \( -2y^{-1/4} \), confirming the original integrand.

Key Concepts

Antiderivative

An antiderivative of a function is essentially a "reverse" process of differentiation. When you find the antiderivative, you uncover the original function from which a given derivative is derived. In mathematical terms, if you have a function \( f(x) \), an antiderivative \( F(x) \) is a function such that \( F'(x) = f(x) \). However, it's important to note that there isn't just one unique antiderivative for a function. Instead, there is a family of functions, as the antiderivative can have an arbitrary constant \( C \) added to it. This is because differentiating the constant results in zero, thus the original function remains unchanged.

For instance, if you consider the function \( f(x) = 2x \), one of its antiderivatives would be \( F(x) = x^2 + C \), where \( C \) is any constant. The indefinite integral \( \int f(x) \, dx \) can be interpreted as the set of all possible antiderivatives of \( f(x) \). When solving problems like the exercise, finding this most general antiderivative involves splitting complex expressions and applying rules like the power rule.

Power Rule for Integration

The power rule for integration is a foundational rule that helps us find antiderivatives easily. It works when integrating terms of the form \( x^n \). The rule states that if you have \( \int x^n \, dx \), the antiderivative will be \( \frac{x^{n+1}}{n+1} + C \), provided that \( n eq -1 \). This rule simplifies the process of integration considerably by providing a formulaic approach.

In the exercise, you apply the power rule individually to each term within the integral \( \int(8 y - \frac{2}{y^{1/4}}) \, dy \). For the first term, \( 8y \), \( n \) is 1, and its antiderivative becomes \( 4y^2 \). For the second term, \( \frac{2}{y^{1/4}} \) is rewritten as \( 2y^{-1/4} \), making it ready for the power rule. By using \( n = -1/4 \), the antiderivative is computed as \( \frac{8}{3}y^{3/4} \).

Remember, applying the power rule requires careful attention to detail in algebraic manipulation and simplifying fractions after integration. Look out for mistakes such as dividing by zero when \( n = -1 \), where different methods apply, such as the natural logarithm integration.

Differentiation Check

After finding an antiderivative, it's crucial to confirm that you've obtained the correct one. A simple way to do this is through the process of differentiation, essentially reversing what you just did. By differentiating your antiderivative and checking whether it matches the original function, you validate your result.

In our exercise, once you have the antiderivative \( 4y^2 - \frac{8}{3}y^{3/4} + C \), differentiate each term:

• The derivative of \( 4y^2 \) is \( 8y \), using the rule \( \frac{d}{dy}(y^n) = ny^{n-1} \).
• The derivative of \( -\frac{8}{3}y^{3/4} \) resolves to \( -2y^{-1/4} \).

Combine these results, and you'll see that the differentiated form matches the original function \( 8y - \frac{2}{y^{1/4}} \), confirming that the antiderivative was correctly found.

By including this differentiation step, you ensure your solution is precise and robust, catching any potential errors made during integration.