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Chapter 5: Problem 8

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int x \sin \left(2 x^{2}\right) d x, \quad u=2 x^{2}$$

Short Answer

Expert verified
The integral is \(-\frac{1}{4} \cos(2x^2) + C\).

Step by step solution

01

Apply the Substitution

In this step, we perform the substitution provided in the problem. Let \( u = 2x^2 \). Next, we need to find \( du \), the differential of \( u \). Calculating the derivative, we get \( du = 4x \, dx \). However, our integral has just \( x \, dx \), so we rearrange to express \( x \, dx \) in terms of \( du \): \( x \, dx = \frac{1}{4} \, du \). Substitute \( u \) and \( du \) into the integral.
02

Substitute into the Integral

Substitute the expressions obtained from the substitution into the integral. Our integral becomes: \( \int x \sin(2x^2) \, dx = \int \sin(u) \cdot \frac{1}{4} \, du \). This simplifies to \( \frac{1}{4} \int \sin(u) \, du \).
03

Integrate the Standard Form

Now, we integrate the standard form \( \frac{1}{4} \int \sin(u) \, du \). The integral of \( \sin(u) \) is \( -\cos(u) \). Therefore, the integral \( \frac{1}{4} \int \sin(u) \, du \) becomes \( -\frac{1}{4} \cos(u) + C \), where \( C \) is the constant of integration.
04

Substitute Back to Original Variable

Since we integrated in terms of \( u \), we need to substitute back to the original variable \( x \). Recall that \( u = 2x^2 \). Therefore, \( -\frac{1}{4} \cos(u) + C \) becomes \( -\frac{1}{4} \cos(2x^2) + C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a powerful technique that simplifies solving indefinite integrals. It involves replacing a part of the integrand with a new variable, making the integral easier to evaluate. This is especially useful when dealing with complex expressions.
To apply this method, follow these steps:
  • Identify a portion of the integral that can be replaced by a single variable, usually denoted as \( u \).
  • Differentiate this new variable \( u \) with respect to \( x \), obtaining \( du \).
  • Express the differential \( dx \) in terms of \( du \) to match the remaining integrand.
  • Substitute back all occurrences of the selected part with \( u \) and \( dx \) with its expression in terms of \( du \).
In our example, \( u = 2x^2 \), and through differentiation, \( du = 4x \, dx \). The substitution transforms the integral into a form that is often more manageable, helping to link the problem back to known integral templates.
Standard Integral Forms
Standard integral forms are everyday integrals with known solutions, often memorized due to their frequent occurrences. These forms allow for quick integration without in-depth calculation.
For common functions like exponential, polynomial, and trigonometric functions, recognizing these patterns can save time and reduce errors.
  • Some standard integrals include \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) and \( \int e^x \, dx = e^x + C \).
  • In trigonometry, the integral \( \int \sin(x) \, dx = -\cos(x) + C \) is a classic example.
In this exercise, after using substitution, the integral \( \frac{1}{4} \int \sin(u) \, du \) needs just the standard integral of \( \sin(u) \), which simplifies the problem drastically, yielding \( -\frac{1}{4} \cos(u) + C \). This changes an otherwise complicated integral into something straightforward.
Trigonometric Integration
Trigonometric integration deals with integrals of trigonometric functions like sine, cosine, and tangent. These are common in calculus, often requiring specific techniques or substitutions.
  • The integral of \( \sin(x) \) is \( -\cos(x) + C \), while \( \int \cos(x) \, dx = \sin(x) + C \).
  • Substitutions are frequent because they simplify the integrals into known forms. For instance, turning the variable into \( u \) simplifies the trigonometric function within the integral.
In the given exercise, the transformation led to an integral involving \( \sin(u) \), directly correlating this problem to standard trig forms. Although trigonometric integration may seem complex, remembering these integral forms and how to employ substitution makes it accessible.

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Most popular questions from this chapter

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