Problem 30 - Thomas Calculus

Question

In Exercises \(17-54\) , find the most general antiderivative or indefinite integral. Check your answers by differentiation. $$ \int\left(\frac{1}{7}-\frac{1}{y^{5 / 4}}\right) d y $$

Step-by-Step Solution

Answer

The antiderivative is \( \frac{1}{7}y - 4y^{-1/4} + C \). Verified by differentiation.

Step 1: Rewrite the Integral

The given integral is \( \int \left( \frac{1}{7} - \frac{1}{y^{5/4}} \right) d y \). We'll start by rewriting it in a form that's easier to integrate. Separate the terms: \( \int \frac{1}{7} \, d y - \int \frac{1}{y^{5/4}} \, d y \). This step helps us manage the expressions individually.

Step 2: Integrate the First Term

Integrate the first term, which is a constant: \( \int \frac{1}{7} \, d y = \frac{1}{7} y + C_1 \). Here, \( C_1 \) is a constant of integration.

Step 3: Integrate the Second Term

For the second integral, \( \int \frac{1}{y^{5/4}} \, d y \), rewrite \( \frac{1}{y^{5/4}} \) as \( y^{-5/4} \). The integral then becomes \( \int y^{-5/4} \, d y \). Use the power rule of integration: \( \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \). Here, \( n = -\frac{5}{4} \), so the integral is \( \frac{y^{-1/4}}{-1/4} + C_2 = -4y^{-1/4} + C_2 \).

Step 4: Sum the Integrals

Now, combine the results of the two integrals: \( \frac{1}{7} y + C_1 - 4y^{-1/4} + C_2 \). Simplify the expression: \( \frac{1}{7} y - 4y^{-1/4} + C \), where \( C = C_1 + C_2 \).

Step 5: Verify by Differentiation

Differentiate the resulting expression to verify. Differentiate \( \frac{1}{7} y \) to get \( \frac{1}{7} \), and differentiate \( -4y^{-1/4} \) to get \( 4 \times \frac{1}{4} \times y^{-5/4} = -y^{-5/4} \). Hence, \( \frac{1}{7} - \frac{1}{y^{5/4}} \) matches the original integrand, confirming the solution is correct.

Key Concepts

Antiderivative

When we talk about antiderivatives, we're essentially searching for a function whose derivative equals the integrand. In simple terms, it's the reverse operation of finding a derivative. This is often referred to as integrating a function.
An antiderivative is not unique since functions can have multiple forms akin, differing by a constant. Therefore, when we find an antiderivative, we always add a "constant of integration," which accounts for all possible shifts vertically. In our exercise, this means both the constants resulting from each integration step are combined to form a general constant.

• Antiderivative is the opposite of differentiation.
• Always include an integration constant, like "+ C", to reflect any possible vertical shifts.

Understanding antiderivatives enables us to solve for unknown functions when given certain conditions, which is crucial in many fields such as physics and engineering.

Indefinite Integral

The indefinite integral of a function represents all its possible antiderivatives. When you see the integral symbol without limits on the top or bottom, that's an indefinite integral. It signifies we're not finding the area under a curve for specific limits, but rather a general formula representing an entire family of functions.
For our example, the indefinite integral is \[\int \left( \frac{1}{7} - \frac{1}{y^{5/4}} \right) dy \]which translates after integration to: \[\frac{1}{7} y - 4y^{-1/4} + C \]Here, "C" symbolizes that family of functions by accommodating their constants of integration.

• No limits in the notation means it’s indefinite.
• Indefinite integrals provide a general solution.

This topic is key for solving real-world problems where conditions or boundaries are not explicitly defined.

Power Rule

The power rule for integration is a handy tool when dealing with functions in polynomial form. It works opposite to the power rule for differentiation, stating that for any term \(y^n\), its integral is \[\int y^n \, dy = \frac{y^{n+1}}{n+1} + C\]assuming \(n eq -1\).
In this exercise, this rule transforms \(y^{-5/4}\) into \[-4y^{-1/4}\]continually simplifying our integration process. This happens by adjusting the exponent and dividing by the new exponent.

• Check the exponent: increase by one.
• Divide by this new exponent.
• Add the constant of integration "C".

Mastering the power rule simplifies integration of polynomial expressions, a cornerstone principle for tackling more complex integrals.

Integration Verification

After executing integration, verifying the results is a crucial step. This helps ensure accuracy and correct interpretation of the original problem. We do this by differentiating the resultant expression and confirming if it matches the original integrand. This step acts as a validation of our solution.
For the exercise provided, once we have the integrated form \[\frac{1}{7} y - 4y^{-1/4} + C\]we differentiate to get back to the integrand \[\frac{1}{7} - \frac{1}{y^{5/4}}\]confirming our integration was performed correctly.

• Differentiation checks our integration work.
• Matching the original integrand confirms correctness.

Regular verification strengthens understanding and boosts confidence in solving integration exercises, ensuring precision throughout the learning process.