Problem 39 - Applied Calculus

Question

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int(2 x-3)^{7} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{(2x-3)^8}{16} + C \).

Step 1: Identify the Inner Function

Recognize that the inner function here is \( u = 2x - 3 \), which is a linear expression inside the parenthesis raised to the power of 7.

Step 2: Calculate the Derivative of the Inner Function

Find the derivative of \( u = 2x - 3 \). The derivative is \( \frac{du}{dx} = 2 \).

Step 3: Express dx in terms of du

Rewrite \( dx \) in terms of \( du \) using the derivative \( \frac{du}{dx} = 2 \). This gives \( dx = \frac{du}{2} \).

Step 4: Substitute u and dx into the Integral

Replace \( 2x - 3 \) with \( u \) and replace \( dx \) with \( \frac{du}{2} \) in the integral: \[ \int (2x-3)^7 \, dx = \int u^7 \, \frac{du}{2} \] which simplifies to \[ \frac{1}{2} \int u^7 \, du \].

Step 5: Integrate with respect to u

Calculate the integral: \( \frac{1}{2} \int u^7 \, du \). The integral of \( u^7 \) is \( \frac{u^8}{8} \), so our expression becomes:\[ \frac{1}{2} \cdot \frac{u^8}{8} + C = \frac{u^8}{16} + C \].

Step 6: Substitute back the original variable

Convert back from \( u \) to \( x \) using \( u = 2x-3 \). Thus:\[ \frac{u^8}{16} + C = \frac{(2x-3)^8}{16} + C \].

Key Concepts

Substitution Method in Integration

The substitution method is a powerful technique that simplifies the process of integration, especially with complex expressions. It involves changing variables to make integration easier. When we see an integral that contains a composition of functions (like \( (2x - 3)^7 \)), we can break it down by substituting part of the function with a single variable \( u \). This effectively transforms a complicated integral into a more manageable form.

• Identify part of the integral as \( u \).
• Calculate the derivative \( \frac{du}{dx} \).
• Replace \( dx \) in terms of \( du \).
• Substitute \( u \) and \( du \) into the integral.
• Simplify the integral and solve it.

These steps convert a challenging problem into a straightforward integral problem that’s easier to solve.

Integration Techniques for Solving Indefinite Integrals

Integration techniques help solve indefinite integrals more efficiently. The solution provided uses substitution to address a polynomial raised to a power. Once we've substituted for \( u \) and adjusted \( dx \) to \( du \), the integral becomes much simpler.
Instead of attempting to integrate \( (2x - 3)^7 \) directly, which is cumbersome, we break it into a simpler problem: \( \frac{1}{2} \int u^7 \, du \). Techniques like recognizing common patterns or using trigonometric identities can also help with different types of integrals.

• Substitution: Simplifies the original problem.
• Integration by Parts: Useful for products of functions.
• Partial Fractions: Breaks rational functions into simpler fractions.

Choosing the right technique depends on the function you aim to integrate. For this exercise, substitution was the optimal choice.

Inner Function Identification

Inner function identification is critical when using the substitution method. Recognizing which part of the integral to substitute is essential for simplifying the integration process.
In our example, the expression \( 2x - 3 \) is the inner function within \( (2x - 3)^7 \). By identifying \( u \) as \( 2x - 3 \), we isolate a troublesome component and rewrite the integral in terms of \( u \).

• Look for compositions of functions (like \( f(g(x)) \)).
• Choose \( u \) such that its derivative \( du \) conveniently appears in the integral.
• Simplify the integral with the substitution and solve.

Identifying the inner function correctly is the key step. It allows the rest of the substitution technique to proceed smoothly, ultimately providing a clear path to solve the integral.