Problem 82 - Applied Calculus

Question

What is wrong with the following use of the substitution \(u=x^{2} ?\) $$ \int e^{x^{2}} d x=\int e^{u} d u=e^{u}+C=e^{x^{2}}+C $$

Step-by-Step Solution

Answer

The substitution isn't valid because it lacks a corresponding \( 2x \) for \( du = 2x \, dx \).

Step 1: Understand the Problem

The problem is asking us to identify what's incorrect about the substitution \( u = x^2 \) in the integral \( \int e^{x^2} \, dx \). It's essential to verify whether the substitution and its differentiation are handled correctly.

Step 2: Check Substitution Derivative

When performing substitution, differentiate the substitution equation \( u = x^2 \). Thus, \( \frac{du}{dx} = 2x \), which implies \( du = 2x \, dx \). There must be a corresponding \( 2x \) in the integral to match \( du \).

Step 3: Analyze Incorrect Substitution Regex

The problem arises because the original integral \( \int e^{x^2} \, dx \) does not contain the term \( 2x \) required to match \( du = 2x \, dx \). Instead, only \( dx \) is available, so the substitution is incorrect as the integral lacks a \( 2x \) to pair with \( du \).

Step 4: Express Error in Integral

The correct substitution process should yield an integral in terms of \( u \) involving \( du \). Since there is no corresponding \( 2x \) in \( \int e^{x^2} \, dx \), the substitution \( \int e^{u} \, du \) is invalid because it doesn't correctly transform the differential \( dx \) to \( du \).

Step 5: Conclusion

To resolve the issue, either modify the integral to include \( 2x \, dx \) or reconsider the substitution approach, as the integral of \( e^{x^2} \) with respect to \( x \) often requires special functions or numerical methods, indicating the error in assuming \( \int e^{u} \, du \) applies directly.

Key Concepts

Integration Techniques

Integration techniques are tools that help us solve integrals, which can be quite challenging, given their varied nature. Not all functions have straightforward antiderivatives, so various methods have developed to address this.

Some common techniques include:

• Substitution: This method simplifies the integral by changing variables.
• Integration by Parts: Useful when the function is a product of two simpler functions.
• Partial Fraction Decomposition: Decomposes complex rational functions into simpler fractions.
• Numerical Integration: Use of computational methods when antiderivatives are not easily accessible.

Each technique is suited for specific types of integrals, and selecting the proper technique is often key to solving an integral efficiently. In the context of the substitution method, identifying when this technique will simplify an integral is crucial, as demonstrated by the mistake in assuming the substitution of \(u = x^2\) directly results in a facile integration process.

Substitution Method

The substitution method in integration is akin to reverse differentiation using the chain rule. This approach simplifies integration by expressing the integrand in terms of a new variable, \(u\), which should ideally lead to a simpler integral.

To correctly apply substitution:

• Select a substitution that simplifies the integrand, usually a function inside another function.
• Differentiate the substitution equation to find \(du\), replacing \(dx\) in the integral.
• Ensure the differential \(dx\) is entirely converted to \(du\).

In the example given, substituting \(u = x^2\) without recognizing that \(du = 2x \, dx\) is essential demonstrates a common pitfall. The integrand lacked the term \(2x\), necessary to align with the substitution, making the attempted substitution invalid. Proper substitution ensures all parts of \(dx\) are expressed in \(du\). If substitution does not simplify the integral, the choice of \(u\) may need reevaluation.

Differential Equations

Differential equations involve functions and their derivatives, often requiring specific integration techniques. When solving these equations, integration plays a critical role.In the context of our exercise, recognizing functions that are naturally solved using substitution can significantly ease the process of solving differential equations involving exponential functions.

Techniques like:

• Separation of Variables: Separating variables to different sides to allow integration.
• Integrating Factor: Used in linear differential equations to facilitate straightforward solving.
• Substitution: Applied to recognize patterns and simplify the structures within the equations.

Understanding these techniques and knowing when each is applicable allows for a streamlined approach to complex differential equations. As witnessed in the attempted substitution of \(u = x^2\), the central challenge often is correctly aligning derivatives and integrals, a skill crucial not just in integration but in the broader scope of differential equations.