Problem 29 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int\left(8 y-\frac{2}{y^{1 / 4}}\right) d y $$

Step-by-Step Solution

Answer

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

Step 1: Break Down the Integral

First, we identify each term in the integral separately: \(\int \left( 8y - \frac{2}{y^{1/4}} \right) \, dy\). The integral consists of two parts: \(8y\) and \(-\frac{2}{y^{1/4}}\). We will find the antiderivative of each term separately.

Step 2: Find the Antiderivative of the First Term

For the first term \(8y\), we apply the power rule for integration: \(\int ay^n \, dy = \frac{a}{n+1}y^{n+1} + C\). Here, \(a=8\) and \(n=1\). The antiderivative is \(\frac{8}{2}y^2 = 4y^2\).

Step 3: Find the Antiderivative of the Second Term

The second term is \(-\frac{2}{y^{1/4}}\), which can be rewritten as \(-2y^{-1/4}\). Using the power rule again \(\int ay^n \, dy = \frac{a}{n+1}y^{n+1} + C\), where \(a=-2\) and \(n=-1/4\), the antiderivative is \(\frac{-2}{3/4}y^{3/4} = -\frac{8}{3}y^{3/4}\).

Step 4: Combine the Antiderivatives

Add the two antiderivatives from Steps 2 and 3: \(4y^2 - \frac{8}{3}y^{3/4}\). Thus, the most general antiderivative is \(F(y) = 4y^2 - \frac{8}{3}y^{3/4} + C\), where \(C\) is an arbitrary constant.

Step 5: Differentiate to Check the Answer

Differentiate \(F(y) = 4y^2 - \frac{8}{3}y^{3/4} + C\) with respect to \(y\). The derivative is \(\frac{d}{dy}(4y^2) - \frac{d}{dy}\left(\frac{8}{3}y^{3/4}\right) = 8y - \frac{2}{y^{1/4}}\), which matches the original function inside the integral. This confirms our antiderivative is correct.

Key Concepts

Antiderivatives

Antiderivatives are the reverse of derivatives. If differentiation takes you from a function to its rate of change, integration (finding antiderivatives) takes you back from that rate of change to the original function. This is the core idea behind indefinite integration. Understanding antiderivatives is crucial because they help determine the general solution to problems involving rates of change.

When you find the antiderivative of a function, you incorporate an arbitrary constant, usually represented as "C". This constant is vital because integration can yield multiple solutions differing only by a constant. For example, if the derivative of a function is zero, the function could be any constant. Hence, the general antiderivative is given by:

• If \( f'(x) = constant \), then \( f(x) = constant + C \)

Recognizing the role of the constant "C" is essential for accurately determining the family of solutions for indefinite integrals.

Power Rule for Integration

The power rule for integration is a handy guideline for finding the antiderivatives of power functions. It states that to integrate a function of the form \( ay^n \), you increase the exponent by one and divide by this new exponent, then multiply by the coefficient "a". The formula is:

• \( \int ay^n \, dy = \frac{a}{n+1}y^{n+1} + C \)

This formula is simple and powerful. It allows you to quickly integrate any term that involves a power of y.

Using the power rule correctly involves some basic algebraic manipulation. For instance, in the exercise, notice how \(-\frac{2}{y^{1/4}}\) is rewritten as \(-2y^{-1/4}\). This manipulation makes it possible to apply the power rule, even with fractional or negative exponents. Such transformations are often required to put a term into a form suitable for straightforward integration.Applying this rule systematically can significantly simplify problems involving polynomials or similar expressions.

Differentiation Check

After finding an antiderivative, the next crucial step is to verify the solution. Differentiation check is the process of differentiating the antiderivative to see if you obtain the original integrand. This step ensures that your integration was carried out correctly.

For example, suppose you have found an antiderivative \( F(y) \), which you suspect is correct. To verify, differentiate \( F(y) \) with respect to \( y \). If this derivative matches the integrand, your solution is confirmed. In our exercise, we derived \( F(y) = 4y^2 - \frac{8}{3}y^{3/4} + C \). Differentiating this gives:

• The derivative of \( 4y^2 \) is \( 8y \)
• The derivative of \( -\frac{8}{3}y^{3/4} \) is \( -\frac{2}{y^{1/4}} \)

Combining these, we get the original integrand, confirming the correctness of our antiderivative determination. This step is an important sanity check and ensures us that the integration process was done accurately.