Problem 47
Question
In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{1}^{5} x e^{-x^{2}} d x $$
Step-by-Step Solution
Verified Answer
\( \frac{1}{2} (e^{-1} - e^{-25}) \).
1Step 1: Choose a Substitution
To simplify the integral, notice that the expression inside the exponential, \(e^{-x^2}\), suggests using substitution. Choose \(u = -x^2\). Then compute the differential: \(du = -2x \, dx\).
2Step 2: Solve for dx in terms of du
From \(du = -2x \, dx\), solve for \(dx\) in terms of \(du\): \(dx = \frac{du}{-2x}\).
3Step 3: Substitute in the Integral
Rewrite the integral using the substitution: replace \(x e^{-x^2} dx\) by \(-\frac{1}{2} e^u du\): \[ \int_{u(1)}^{u(5)} -\frac{1}{2} e^u \, du \].
4Step 4: Find New Limits of Integration
The original limits are for \(x\). Calculate the new limits for \(u\): When \(x = 1, u = -1^2 = -1\), and when \(x = 5, u = -5^2 = -25\).
5Step 5: Evaluate the New Integral
Evaluate the new integral: \[ \int_{-1}^{-25} -\frac{1}{2} e^u \, du = -\frac{1}{2} \left[ e^u \right]_{-1}^{-25} \].
6Step 6: Calculate Result after Integration
Substitute the upper and lower limits: \(-\frac{1}{2} (e^{-25} - e^{-1}) = \frac{1}{2} (e^{-1} - e^{-25})\).
7Step 7: Final Simplification
Calculate the numerical value if necessary: Since \(e^{-1}\) and \(e^{-25}\) can be complex, the expression \(\frac{1}{2} (e^{-1} - e^{-25})\) is the simplified exact form.
Key Concepts
Substitution Method ExplainedUnderstanding the Limit of IntegrationDifferential Calculus and Its Role
Substitution Method Explained
The substitution method is a vital technique in calculus used to simplify complex integrals, particularly when dealing with functions that have compositions or products that make direct integration challenging. The essence of this method is to transform the integral into a simpler form using a substitution that eases the integration process.
In this case, we examined the integral \( \int_{1}^{5} x e^{-x^{2}} \, dx \). We noticed that the expression \( e^{-x^2} \) suggests a substitution due to its complexity. We set \( u = -x^2 \), helping convert the integral into a more manageable form. This change of variables is crucial as it allows us to focus on integrating a simpler expression with respect to the new variable \( u \).
The differential \( du \) is also derived from the substitution, providing a way to express \( dx \) in terms of \( du \). All components of the original integral are then expressed using the new variable. This simplification turns the original difficult integral into a straightforward problem to solve.
In this case, we examined the integral \( \int_{1}^{5} x e^{-x^{2}} \, dx \). We noticed that the expression \( e^{-x^2} \) suggests a substitution due to its complexity. We set \( u = -x^2 \), helping convert the integral into a more manageable form. This change of variables is crucial as it allows us to focus on integrating a simpler expression with respect to the new variable \( u \).
The differential \( du \) is also derived from the substitution, providing a way to express \( dx \) in terms of \( du \). All components of the original integral are then expressed using the new variable. This simplification turns the original difficult integral into a straightforward problem to solve.
Understanding the Limit of Integration
When transforming an integral via substitution, it's key not only to change the variables but also the limits of integration. The limits reflect the range of the variable we are integrating across. Initially, we had limits \( x = 1 \) and \( x = 5 \).
With our chosen substitution, \( u = -x^2 \), the new limits of integration also change. When \( x = 1 \), this maps to \( u = -1^2 = -1 \). Similarly, when \( x = 5 \), \( u = -5^2 = -25 \). These calculations ensure that our integral covers the correct range in the changed variable space.
With our chosen substitution, \( u = -x^2 \), the new limits of integration also change. When \( x = 1 \), this maps to \( u = -1^2 = -1 \). Similarly, when \( x = 5 \), \( u = -5^2 = -25 \). These calculations ensure that our integral covers the correct range in the changed variable space.
- The original bounds need to be adjusted in line with the substitution to ensure continuity.
- Preserving these limits ensures that the definite integral calculates the exact area under the curve in the new variable.
Differential Calculus and Its Role
Differential calculus provides tools for understanding and computing the rates of change and slopes of curves. When we use substitution in integration, differential calculus is at play, specifically through the computation of differentials.
In our exercise, we determined \( du = -2x \, dx \) which connects to our substitution \( u = -x^2 \). This expression of \( du \) allows us to replace the \( dx \) in the integral with terms of \( du \), effectively changing the integrand to match the new variable.
In our exercise, we determined \( du = -2x \, dx \) which connects to our substitution \( u = -x^2 \). This expression of \( du \) allows us to replace the \( dx \) in the integral with terms of \( du \), effectively changing the integrand to match the new variable.
- The role of differential calculus here is to facilitate the transformation of variables smoothly from \( x \) to \( u \).
- This process highlights how calculus can break down complex functions into manageable parts by analyzing infinitesimally small changes.
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