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Chapter 5: Problem 12

Show that each integral cannot be found by our substitution formulas. $$ \int e^{x^{3}} x^{4} d x $$

Short Answer

Expert verified
Substitution methods don't work for this integral.

Step by step solution

01

Identify the integral structure

We have the integral \( \int e^{x^{3}} x^{4} \, dx \). We begin by examining if it fits a standard substitution pattern, such as \( u = x^3 \) or \( u = e^{x^3} \).
02

Attempt suitable substitution

Try \( u = x^3 \), so that \( du = 3x^2 \, dx \) and \( x^4 \, dx = \frac{x^2}{3} \, du \). This does not match the given integral because \( x^4 \) must be expressible in terms of \( du \) to substitute properly.
03

Explore alternative substitutions

Consider \( u = e^{x^3} \), leading to \( du = 3x^2 e^{x^3} \, dx \), which simplifies to \( x^4 \, dx = \frac{x^2}{3} \, du \). Again, this does not help, as \( x^4 \) cannot be neatly expressed in terms of \( du \).
04

Confirm mismatch with substitution patterns

Since neither substitution simplifies the integral into a standard form that allows separation of \( x^4 \, dx \) entirely in terms of \( du \), we confirm that basic substitution methods do not solve this integral.

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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 technique in calculus used to simplify integrals by changing variables. It can make complex integrals much easier to solve by transforming them into a more manageable form. The core idea is to substitute a new variable for a part of the integral's expression.
  • Choose a substitution: Typically, you look for a function within the integral that, when differentiated, appears elsewhere in the integrand.
  • Change variables: Re-express the integral in terms of the new variable.
  • Back-substitute: Solve the integral in terms of the new variable, then replace it back with the original variable.
For the integral \( \int e^{x^{3}} x^{4} \, dx \), common substitutions do not simplify the problem. By trying \( u = x^3 \) or \( u = e^{x^3} \), the integral cannot wholly be expressed in terms of \( du \). This indicates that more advanced techniques may be needed as the substitution method alone is insufficient here.
Non-standard Integrals
When dealing with calculus problems, you may encounter non-standard integrals. These are integrals that do not fit well into known categories for easy resolution using standard calculus techniques like substitution or integration by parts.Such integrals often arise when:
  • The integrand has no apparent antiderivative in terms of elementary functions.
  • Attempts at variable substitution, like in our example, do not simplify the integral.
  • Established techniques yield no meaningful simplification or are incompatible with the integrand structure.
In the case of \( \int e^{x^{3}} x^{4} \, dx \), attempts to apply substitution methods fail because the integrand's complexity exceeds typical substitution capabilities. A more advanced approach may involve special functions or numerical integration methods for evaluation.
Calculus Problem Solving
Solving calculus problems, especially when encountering challenging integrals, involves a strategic approach. Here are some steps to tackle these issues:
  • Identify integral patterns: Look for common structures that match known integral forms.
  • Attempt substitutions: Experiment with different substitution methods to simplify the expression.
  • Review assumptions: Check assumptions made during substitution to ensure validity.
In the case of problematic integrals, like \( \int e^{x^{3}} x^{4} \, dx \), it's essential to recognize when traditional methods are inadequate. Examine the possibility of using advanced techniques such as:
  • Partial fractions
  • Integration by parts
  • Special functions that extend beyond elementary functions, like the Gamma or Error functions
Lastly, consider numerical approaches when analytic solutions aren't feasible, ensuring a thorough exploration of tools in the calculus toolbox.

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Most popular questions from this chapter

\(75-76 .\)A factory is discharging pollution into a lake at the rate of \(r(t)\) tons per year given below, where \(t\) is the number of years that the factory has been in operation. Find the total amount of pollution discharged during the first 3 years of operation. $$ r(t)=\frac{t}{t^{2}+1} $$

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