Problem 13 - Calculus Single Variable

Question

Calculate the indefinite integral. $$ \int\left(x^{-7 / 3}-4 x^{-2 / 3}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( -\frac{3}{4}x^{-4/3} - 12x^{1/3} + C \).

Step 1: Understanding the Terms

The integral given is \( \int \left( x^{-7/3} - 4x^{-2/3} \right) \, dx \). We need to integrate each term separately: \( x^{-7/3} \) and \( -4x^{-2/3} \).

Step 2: Integrate the First Term

Use the power rule for integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n eq -1 \). Here, for \( x^{-7/3} \), \( n = -7/3 \), so the integral becomes: \[ \int x^{-7/3} \, dx = \frac{x^{-7/3+1}}{-7/3+1} = \frac{x^{-4/3}}{-4/3} = -\frac{3}{4}x^{-4/3} \]

Step 3: Integrate the Second Term

Similarly, for the term \( -4x^{-2/3} \), apply the power rule: \[ \int (-4)x^{-2/3} \, dx = -4 \int x^{-2/3} \, dx = -4 \left( \frac{x^{-2/3+1}}{-2/3+1} \right) = -4 \left( \frac{x^{1/3}}{1/3} \right) = -4 \times 3x^{1/3} = -12x^{1/3} \]

Step 4: Combine the Integrals

Combine the two results obtained from integrating each term: \[ -\frac{3}{4}x^{-4/3} - 12x^{1/3} + C \] where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

When we talk about integration, one of the essential rules to remember is the Power Rule for Integration. This rule simplifies the process and is key in dealing with expressions with powers of variables. The Power Rule states that for a term in the form of \( x^n \) (where \( n eq -1 \)), the integral is given by:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Here, you raise the power by one and then divide by the new power. This process effectively "undoes" differentiation of power terms.
In our example, we applied this rule to each term: \( x^{-7/3} \) and \( -4x^{-2/3} \). We added 1 to the exponent in each case and divided by the new exponent.
It's important to be comfortable with fractions here, and to remember that if the power is exactly \( -1 \), different rules, involving logarithms, would apply.

Integration of Monomials

Monomials are single term expressions like \( ax^n \). Integrating them is a process akin to our integration of simple power expressions, but we have a coefficient to consider as well.

• In the original problem, we identified two monomials: \( x^{-7/3} \) and \( -4x^{-2/3} \).
• For \( -4x^{-2/3} \), we first factor out the constant \(-4\), then apply the Power Rule to \( x^{-2/3} \).

The integration of a monomial begins with the same principles as applying the Power Rule, but you must remember to include the coefficient in your final answer.
This is why, for \( -4 \int x^{-2/3} \, dx \), once we find \( \frac{x^{1/3}}{1/3} \), multiplying by \(-4\) gives \(-12x^{1/3}\). This step highlights the importance of handling constants separately.

Constant of Integration

The constant of integration, denoted \( C \), is a crucial element in indefinite integration. When dealing with indefinite integrals, we don't have limits of integration, which means that there could be infinitely many answers differing by a constant.
The constant of integration represents this undetermined constant. Without it, you would suggest that there's only one possible antiderivative, which isn't true.
In our step-by-step solution, once we've calculated the antiderivative of each term, we added \( C \) to represent the family of all antiderivatives.

• While it may seem like a small addition, \( C \) is essential for conveying the general solution.
• It reflects the idea that for any given position of the function's graph vertically, you might find another similar looking function, shifted up or down, still satisfying the differential equation derived from the derivative.

Never forget to add \( C \) when solving indefinite integrals; it's a critical part of the complete solution!