Problem 21 - Applied Calculus

Question

Find each indefinite integral. \(\int\left(10 \sqrt[3]{t^{2}}+\frac{1}{\sqrt[3]{t^{2}}}\right) d t\)

Step-by-Step Solution

Answer

The integral is \(6t^{\frac{5}{3}} + 3t^{\frac{1}{3}} + C\).

Step 1: Break Down the Integral

Identify each component of the integral that needs to be integrated separately: \(10 \sqrt[3]{t^2}\) and \(\frac{1}{\sqrt[3]{t^2}}\). Rewrite these components using power notation.

Step 2: Rewrite the Terms Using Exponents

The term \(\sqrt[3]{t^2}\) can be rewritten as \(t^{\frac{2}{3}}\), and \(\frac{1}{\sqrt[3]{t^2}}\) can be rewritten as \(t^{-\frac{2}{3}}\). Now the integral becomes \(\int \left(10t^{\frac{2}{3}} + t^{-\frac{2}{3}}\right) dt\).

Step 3: Integrate Each Term Separately

Use the power rule for integration \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\) for each term. For \(10t^{\frac{2}{3}}\), integrate to get \(10 \cdot \frac{t^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} = 10 \cdot \frac{t^{\frac{5}{3}}}{\frac{5}{3}} = 6t^{\frac{5}{3}}\). For \(t^{-\frac{2}{3}}\), integrate to get \(\frac{t^{-\frac{2}{3} + 1}}{-\frac{2}{3} + 1} = \frac{t^{\frac{1}{3}}}{\frac{1}{3}} = 3t^{\frac{1}{3}}\).

Step 4: Combine the Results

Add the integrals of each term to get the complete indefinite integral. Combine the results: \(6t^{\frac{5}{3}} + 3t^{\frac{1}{3}} + C\). The constant \(C\) represents the indefinite nature of the integration.

Key Concepts

Indefinite Integral

The concept of an indefinite integral in calculus can initially seem complex, but it's quite approachable once you understand its fundamentals. An indefinite integral, often denoted as \( \int f(x) \, dx \), represents the collection of all antiderivatives of a function \( f(x) \). Unlike a definite integral, it does not have upper and lower limits and results in a general expression plus a constant \( C \).

That constant \( C \) is crucial because when differentiating the original function, any constant would disappear, due to the derivative of a constant being zero. Thus, all indefinite integrals must include this constant to be accurate. The main objective is to "undo" the differentiation process, finding the original function or functions that could have produced \( f(x) \) when differentiated. In this regard, the process is often referred to as "antidifferentiation."

Whenever you encounter an integral symbol without limits, remember that you're searching for the family of functions that could yield the integrand upon differentiation.

Power Rule for Integration

One of the most frequently used techniques in integration, especially when dealing with polynomials, is the Power Rule for Integration. The Power Rule simplifies solving integrals of the form \( \int x^n \, dx \).

The rule states: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n eq -1 \). The rule helps transform the power of the variable \( x \) by increasing it by one and then dividing by this new power. This pattern makes it incredibly straightforward to integrate functions that are simple polynomials or can be transformed into such.

In practical applications, like in the provided exercise, using the Power Rule allows one to handle each term of the integrand separately and later combine all integrated terms. The key is to carefully transform any fractions or roots into power form before applying the rule. Thus, it is often paired with algebraic manipulations to set the problem in a workable form, making integration more manageable.

Integration Techniques

Integration involves various techniques, and selecting the correct one is vital to solving integral problems efficiently. The primary techniques include substitution, integration by parts, partial fractions, and the straightforward approach illustrated in this exercise, often called direct integration.

Direct integration is highly effective when the integrand is a simple polynomial or can be rewritten as such through algebraic manipulation. Rewriting expressions using exponents, as seen with roots in the given exercise, allows the use of elementary rules like the Power Rule.

When confronted with more complex integrands, substitution might come in handy, especially when an integral can be expressed by recognizing a component that is the derivative of another component. This technique makes it easier to simplify the integral.

Every technique has its proper context and application, and recognizing which to apply comes with practice and familiarity with different functional forms. The goal of learning these techniques is to provide a diverse toolkit that can address a wide range of integration problems with confidence and precision.