Problem 10 - Technical Mathematics with Calculus

Question

Integrate $$\int 5 x \cos 2 x^{2} d x$$

Step-by-Step Solution

Answer

\frac{5}{4} \text{sin}(2x^2) + C

Step 1: Recognize the Integration Pattern

Identify the integral as a product of functions that suggests the use of the substitution method. In this case, it is the product of a polynomial function, \(x\), and a trigonometric function, \(\cos(2x^2)\).

Step 2: Choose a Substitution

Make a substitution to simplify the integral. Let \(u = 2x^2\). Then, compute the derivative of \(u\) with respect to \(x\) to find \(du\).

Step 3: Find du

Differentiate \(u = 2x^2\) with respect to \(x\) to get \(du = 4x dx\). We need to find a match for the term \(5x dx\) in the integral, so we solve for \(x dx\) in terms of \(du\): \(\frac{1}{4}du = x dx\).

Step 4: Adjust the Integral for Substitution

Express \(5x dx\) in terms of \(du\), by multiplying both sides of the equation \(\frac{1}{4}du = x dx\) by \(5\): \(\frac{5}{4}du = 5x dx\). Now we can replace \(5x dx\) with \(\frac{5}{4}du\) in the integral.

Step 5: Substitute and Integrate

Substitute \(2x^2\) with \(u\) and \(5x dx\) with \(\frac{5}{4}du\) in the integral. The integral now becomes \(\frac{5}{4} \int \cos u du\). Integrate \(\cos u\) with respect to \(u\) to get \(\text{sin} u\).

Step 6: Apply the Constant Multiple Rule

Using the constant multiple rule, take the constant \(\frac{5}{4}\) out of the integral. Now perform the integral with the constant outside: \(\frac{5}{4} \int \cos u du = \frac{5}{4} \text{sin} u\).

Step 7: Back-Substitute to Original Variable

After finding the indefinite integral in terms of \(u\), substitute back the original variable \(x\) using the initial substitution \(u = 2x^2\). This gives us the final answer: \(\frac{5}{4} \text{sin}(2x^2) + C\), where \(C\) is the constant of integration.

Key Concepts

Integration Techniques

Understanding different integration techniques is crucial for solving a wide variety of calculus problems. These techniques allow us to transform complex integrals into simpler forms that can be easily evaluated. One common method is the power rule for integration, suitable for polynomials. Another vital technique involves integrating exponential functions, where the base of the exponent plays a significant role.

For integrals involving products of functions or compositions, like the product of a polynomial and a trigonometric function, techniques like integration by parts and u-substitution are often applied. The goal with these methods is to rewrite the integral into a form that is more straightforward to integrate, either by reducing the integrand to a basic form we can integrate directly or by transforming the integral into a simpler equivalent integral.

Trigonometric Integrals

Dealing with trigonometric integrals involves integrating functions that contain trigonometric expressions. This task can sometimes be straightforward, such as integrating basic trigonometric functions like \( \sin x \) or \( \cos x \). However, integration becomes more challenging when the trigonometric functions are multiplied by other types of functions, such as polynomials, or when they are involved in more complex expressions.

Standard Trigonometric Integrals

• \( \int \sin x \, dx = -\cos x + C \)
• \( \int \cos x \, dx = \sin x + C \)
• \( \int \tan x \, dx = -\ln |\cos x| + C \)
• \( \int \cot x \, dx = \ln |\sin x| + C \)
• \( \int \sec x \, dx = \ln |\sec x + \tan x| + C \)
• \( \int \csc x \, dx = -\ln |\csc x + \cot x| + C \)

For products of trigonometric and other functions, we can often make the integration process more manageable by using substitution methods or trigonometric identities to rewrite the integrals into a simpler form.

U-Substitution

The method of u-substitution, often referred to as integration by substitution, is a powerful tool that can simplify many integrals. It is essentially the reverse process of the chain rule used in differentiation. The main idea is to choose a new variable \(u\) that simplifies the integral when substituted for the existing variable.

To apply u-substitution, identify a part of the integral that when differentiated, can closely resemble the rest of the integrand. After choosing an appropriate \(u\), differentiate it to determine \(du\) and then express \(dx\) or some portion of the integrand in terms of \(du\). This substitution makes the integral easier to evaluate, often reducing it to a basic form.

Steps for U-Substitution

1. Choose \(u\) to simplify the integral.
2. Differentiate \(u\) and solve for \(dx\).
3. Substitute \(u\) and \(dx\) into the integral.
4. Integrate with respect to \(u\).
5. Substitute back to the original variable if necessary.

In practice, it's not always immediately clear what to choose for \(u\), and the choice can significantly affect the simplicity of the process. Hence, practicing a variety of problems is the key to mastering u-substitution. It's especially useful when faced with a product of a polynomial and a trigonometric function, as seen in the given exercise.