Problem 7 - Calculus for Biology and Medicine

Question

In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int 7 x \sin \left(4 x^{2}\right) d x \text { , with } u=4 x^{2} $$

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{7}{8} \cos(4x^2) + C\)."

Step 1: Identify the substitution variable

We are given the substitution variable as \( u = 4x^2 \). This will help us simplify the integral.

Step 2: Differentiate to find du

Differentiate \( u = 4x^2 \) with respect to \( x \) to find \( du \). We get \( du = 8x \, dx \).

Step 3: Solve for dx in terms of du

Rearrange \( du = 8x \, dx \) to express \( dx \) in terms of \( du \) and \( x \). So, \( dx = \frac{du}{8x} \).

Step 4: Substitute variables in the integral

Substitute \( u = 4x^2 \) and \( dx = \frac{du}{8x} \) into the original integral. The integral becomes \( \int 7x \sin(u) \cdot \frac{du}{8x} \).

Step 5: Simplify the integral

Cancel out \( x \) from the numerator and denominator, giving us \( \int \frac{7}{8} \sin(u) \, du \). Factor out the constant: \( \frac{7}{8} \int \sin(u) \, du \).

Step 6: Integrate with respect to u

The integral \( \int \sin(u) \, du \) is \(-\cos(u) + C\). Thus, we have \( \frac{7}{8} \left(-\cos(u) \right) + C \).

Step 7: Substitute back the original variable

Replace \( u \) with \( 4x^2 \) to return to the original variable. The result is \(-\frac{7}{8} \cos(4x^2) + C\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in simplifying and evaluating indefinite integrals. It helps in transforming a complex integral into one that is easier to work with. When applying this method, we replace a specific part of the integral with a new variable, usually denoted as \( u \). This variable is chosen strategically to simplify the integral's expression.
Here’s how it works:

• Choose a substitution: Look for a function in the integrand that, when replaced by \( u \), simplifies the integral significantly.
• Find \( du \): Once \( u \) is chosen, differentiate it to find \( du \). This will involve taking the derivative of \( u \) with respect to \( x \) and representing the differential \( dx \) in terms of \( du \).
• Transform the integral: Replace the original integral's parts with the new variable \( u \) and express \( dx \) using \( du \). This substitution often results in a simpler integral that's easier to solve.

By effectively implementing the substitution method, complex integrals become approachable, allowing for seamless calculations.

Integration by Substitution

Integration by substitution is essentially the reverse chain rule for solving integrals. It is specifically useful for integrals where a component's derivative appears elsewhere in the integral.
To illustrate, let's revisit our integral: \( \int 7x \sin(4x^2) \, dx \) with substitution \( u = 4x^2 \).

• Differentiate \( u \): Start by differentiating \( u = 4x^2 \) to determine \( du \), giving \( du = 8x \, dx \).
• Express \( dx \) in terms of \( du \): Rearrange to find \( dx = \frac{du}{8x} \).
• Substitute in the integral: Replace \( 4x^2 \) with \( u \) and \( dx \) with \( \frac{du}{8x} \) in the integral, leading to \( \int 7x \sin(u) \cdot \frac{du}{8x} \).
• Simplify: Simplify by canceling and reorganizing, resulting in a manageable form \( \frac{7}{8} \int \sin(u) \, du \).

Finally, integrate with respect to \( u \) and substitute back the original variable to achieve the solution. The elegance of substitution shines by transforming a daunting integral into a straightforward process.

Trigonometric Integration

Trigonometric integration often involves integrating functions with trigonometric expressions. These can be considerably tricky due to their oscillatory nature. However, using substitution, such integrals become conquerable.
In our exercise, the integral \( \int 7x \sin(4x^2) \, dx \) becomes simpler by focusing on the trigonometric part \( \sin(4x^2) \).

• Integrate sine function: After substitution, the integral \( \int \sin(u) \, du \) is straightforward. The result of this integration is \(-\cos(u) + C\).
• Focus on familiarity: Sine and cosine integrals have well-known results, which helps in quick evaluations once substitution clears the complexity.
• Revert and solve: Once integrated, return to the original variable using the substitution made initially. In this scenario, replace \( u \) with \( 4x^2 \).

This streamlined approach to trigonometric integration through substitution highlights the method's power in resolving otherwise intricate problems involving sines and cosines. By substituting strategically and integrating, one can efficiently deal with trigonometric integrals, turning them from challenging tasks into routine exercises.