Problem 28 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \sqrt[3]{x^{3}-8} x d x $$

Step-by-Step Solution

Answer

The integral cannot be solved by substitution formulas.

Step 1: Identify the Inner Function

The function inside the cube root, \( x^3 - 8 \), is a candidate for substitution. Let \( u = x^3 - 8 \).

Step 2: Compute the Derivative of the Inner Function

Find the derivative of \( u \), which is \( du = 3x^2 \, dx \). This surrogate derivative will help us later replace \( dx \) or other components of the integrand.

Step 3: Express \( dx \) in terms of \( du \)

Solve \( 3x^2 \, dx = du \) to express \( dx \) as \( \frac{du}{3x^2} = dx \).

Step 4: Attempt Substitution

Substitute \( u \) into the integrand, replacing \( \sqrt[3]{x^3 - 8} \) with \( u^{1/3} \) and \( dx \) with \( \frac{du}{3x^2} \). But we notice that \( x \) and \( x^2 \) still remain in the integral, and these are not expressible in terms of \( u \) easily.

Step 5: Evaluate Substitution Possibility

To express \( x \) in terms of \( u \), solve \( u = x^3 - 8 \) for \( x \) to obtain \( x = \sqrt[3]{u + 8} \), but this complicates the integral beyond straightforward substitution, indicating substitution isn't easing the integral.

Step 6: Conclusion

Since the substitution does not eliminate \( x \) entirely from the integrand without overly complex manipulation, state that a straightforward substitution method is not applicable to this integral.

Key Concepts

Substitution Method

The substitution method is like a puzzle piece in calculus. It's a clever technique used to simplify complex integrals by transforming them into more manageable forms. The main idea is to substitute a part of the integrand with a single variable, usually denoted by \( u \), which makes the integration process smoother.
By doing this, we can often turn a complicated problem into a much simpler one by reducing its complexity. When we substitute, we're essentially converting the integral into a new form, involving only the variable \( u \) and terms of \( du \). This can often reveal hidden patterns or easier paths to solve integrals.
However, not every integral is suitable for substitution. If the original variables cannot be completely replaced by the new variable \( u \) and its derivative, substitution may not simplify the problem enough for easy integration. It's like trying to fit a piece where it doesn't belong in a puzzle—it just won’t work.

Inner Function

In the context of substitution, the inner function refers to a distinct part of a composite function that we identify for substitution. Consider it as the 'heart' of the integrand.
For example, in the integral \( \int \sqrt[3]{x^3-8} x \, dx \), the expression \( x^3 - 8 \) is our inner function. It's usually tucked inside a larger function and can be isolated for simplification.
By isolating the inner function, we can designate it as a new variable, say \( u \). This new representation helps to simplify the original integral problem into one involving \( u \), which is theoretically easier to integrate.

• Identifying the inner function is a crucial first step in the substitution method, acting as the gateway to simplification.
• Once identified, we compute its derivative to express changes in terms of the new variable.

Effectively managing the transition from the inner function to the new variable \( u \) is essential for solving through substitution.

Derivative

When doing substitution in integrals, the derivative of the inner function plays a pivotal role. The derivative represents the rate of change of the function, and in substitution, it's crucial for adjusting the differentials in the integrand.
Once you've identified the inner function, like \( x^3 - 8 \), you take its derivative to find \( du \). This gives us \( du = 3x^2 \, dx \).
Finding this derivative helps convert the variables in the original integral. It acts like a bridge, allowing us to replace \( dx \) with \( du \) and facilitating the transition from one set of variables to another.

• We solve for \( dx \) using the derivative: \( \frac{du}{3x^2} = dx \).
• This helps in re-expressing the integrand in terms of \( du \).

However, it's important to note that sometimes the remaining parts of the integral may not fully convert to the new variables, complicating the process and indicating that substitution might not work as intended.

Integrand

The integrand is the expression inside the integral sign that you're trying to find the integral of. It's like the main ingredient in a recipe. Without it, there is no integral to solve!
In our problem, the integrand is \( \sqrt[3]{x^3 - 8} x \). The complexity of this expression is why we consider using the substitution method.
The goal with substitution is to reframe the integrand in terms of the new variable \( u \). This involves identifying a part of the integrand that can be simplified using substitution, along with its derivative, to express the entire integrand differently.

• Transforming the integrand is key to solving the integral more easily.
• In an ideal case, the transformation via substitution results in a simpler integrand that's straightforward to integrate.

Though sometimes, as in our exercise, not all integrands can be easily transformed, showing the limits of certain integration techniques.