Problem 24 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int\left(x^{2}-4\right)^{6} 3 x d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{3}{14} (x^2 - 4)^7 + C \).

Step 1: Identify Substitution Variables

To use the substitution method, we identify a potential substitution. A common choice is to set the inner function of a composition as the substitution. Thus, let \( u = x^2 - 4 \).

Step 2: Differentiate the Substitution

Differentiate the substitution relation \( u = x^2 - 4 \) to find \( du \) in terms of \( dx \). This gives us \( du = 2x \, dx \).

Step 3: Solve for dx

We need to express \( dx \) in terms of \( du \) and variables from the integral. Rearrange \( du = 2x \, dx \) to get \( dx = \frac{du}{2x} \).

Step 4: Substitute into the Integral

Substitute \( u \) and \( dx \) into the integral. The integral \( \int (x^2 - 4)^6 \, 3x \, dx \) becomes \( \int u^6 \, \frac{3}{2} \, du \) because \( 3x \, dx = \frac{3}{2} \, du \).

Step 5: Integrate with Respect to u

Integrate \( \int u^6 \, \frac{3}{2} \, du \). This evaluates to \( \frac{3}{2} \cdot \frac{u^7}{7} + C = \frac{3}{14} u^7 + C \).

Step 6: Back-Substitute for x

Substitute \( x^2 - 4 \) back in for \( u \). This yields \( \frac{3}{14} (x^2 - 4)^7 + C \).

Step 7: State the Final Answer

The indefinite integral evaluates to \( \frac{3}{14} (x^2 - 4)^7 + C \), where \( C \) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus used to simplify the process of finding integrals. This approach involves replacing a complicated part of an integral with a simpler variable.

Here's how it works: you identify an inner function within the integral, typically part of a composite function, and set this inner function equal to a new variable, often denoted as \( u \). For the given integral, \( \int (x^2 - 4)^6 \, 3x \, dx \), we chose \( u = x^2 - 4 \). With this substitution, the integral becomes simpler to handle once we also determine how \( dx \) is related to \( du \).

In practice, after choosing \( u \), the next step is to differentiate \( u \) with respect to \( x \) to obtain \( du \). In our example, differentiating \( u = x^2 - 4 \) results in \( du = 2x \, dx \). This allows us to express \( dx \) in terms of \( du \) and frequently simplifies the entire integral into a more manageable form.

Calculus Problem Solving

When solving calculus problems, such as finding an indefinite integral, it's important to systematically approach the solution.

The first step is often to assess the integral and decide if a substitution or another technique is appropriate. This requires recognizing the composition of functions or spotting standard integral forms.

After deciding on the substitution method, go through the process carefully: choose a substitution that simplifies the integral, compute its differential, rearrange to express \( dx \) in terms of \( du \), and then substitute these into the original integral expression.

Once the integral in terms of \( u \) is obtained, carry out the integration as you would for any simple integral.

• Check each step to avoid small errors that can complicate the final result.
• Don't forget to back-substitute the original variable expression once the integral is evaluated.
• Always include a constant of integration, \( C \), since indefinite integrals represent any family of functions differing by a constant.

Integration Techniques

Integration techniques are essential tools in calculus, enabling us to solve a wide variety of integration problems.

Some common techniques include:

• Substitution: As applied here, it simplifies the integration by changing variables.
• Integration by Parts: Useful for products of functions, applying the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fractions: Decomposes a rational function into simpler fractions that are easier to integrate.
• Trigonometric Integrals/Identities: Useful when dealing with trigonometric functions.

Understanding when and how to apply these techniques is crucial for effective calculus problem solving. Different problems require different strategies, often involving a creative choice of technique.

Definite and Indefinite Integrals

Integrals in calculus are divided into two main types: definite and indefinite integrals.

Indefinite integrals involve finding a function that represents the antiderivative of another function. This process does not produce a finite number, but rather a function plus a constant: \( F(x) + C \). It essentially "reverses" differentiation.

• Example: If the derivative of \( F(x) \) is \( f(x) \), then the indefinite integral of \( f(x) \) is \( F(x) + C \).
• The constant \( C \) represents an infinite number of potential antiderivatives caused by the lack of boundary constraints.

Definite integrals, in contrast, calculate the area under the curve of a function between two specified limits, \( a \) and \( b \). The result is a fixed number, not a function.

• Given by \( \int_{a}^{b} f(x) \, dx \), this type of integration assumes limits of integration and isn't influenced by any additional constant.

Each type serves its purpose, with indefinite integrals used for general solutions and definite integrals providing actual numerical values within specified boundaries.