Problem 62
Question
Use an appropriate substitution followed by integration by parts to evaluate $$ \int x^{3} e^{-x^{2} / 2} d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to \(-x^2 e^{-x^2/2} + 2e^{-x^2/2} + C\).
1Step 1: Identify Substitution
Notice that the integrand contains the expression \(-x^2/2\). It is often useful to substitute \(u = -x^2/2\) because it simplifies the exponential part of the integrand.
2Step 2: Calculate Differential
From the substitution \(u = -x^2/2\), derive the differential: \(du = -x dx\) or \(x dx = -du\).
3Step 3: Rewrite Integral
Substitute the expressions into the integral: \(\int x^{3} e^{-x^{2} / 2} \, dx = \int x^2 \cdot x \, e^u \, dx = -\int x^2 e^u \, du\).
4Step 4: Simplify Further
Since \(x^2 = -2u\) as derived from \(u = -x^2/2\), substitute to simplify the integral: \(-\int (-2u) e^u \, du = 2\int u e^u \, du\).
5Step 5: Integration by Parts Setup
Use integration by parts for \(\int u e^u \, du\). Let \(v = u\) and \(dw = e^u du\); then \(dv = du\) and \(w = e^u\).
6Step 6: Integration by Parts Formula
Apply integration by parts formula \(\int v \, dw = vw - \int w \, dv\), resulting in: \(\int u e^u \, du = u e^u - \int e^u \, du\).
7Step 7: Solve Integral by Parts
Evaluating \(\int e^u \, du = e^u\), the integral becomes \(u e^u - e^u\). Restoring the factors, we have \(2 (u e^u - e^u)\).
8Step 8: Substitute Back and Simplify
We substitute back \(u = -x^2/2\) into the result: \(2((-x^2/2)e^{-x^2/2} - e^{-x^2/2}) = -x^2 e^{-x^2/2} + 2e^{-x^2/2}\).
9Step 9: Final Integral Solution
The evaluated integral is written as \(-x^2 e^{-x^2/2} + 2e^{-x^2/2} + C\), where \(C\) is the constant of integration.
Key Concepts
Substitution MethodDefinite IntegralsExponential Functions
Substitution Method
The substitution method is a powerful technique in integration, particularly useful when dealing with complex expressions such as those involving exponential functions. The idea is to simplify the integrand by introducing a new variable, called the substitution. In this exercise, we noticed the term \(-x^2/2\) within the exponential function, and by setting \(u = -x^2/2\), we can convert a seemingly complicated integral into a friendlier one.
When performing substitution, it is crucial to also adjust the differential. For our substitution, we derived: \( du = -x \, dx \, \text{or} \, x \, dx = -du\). This transformation allows us to rewrite the integral in terms of \(u\): \(-\int x^2 \, e^u \, du\).
Thus, the substitution method simplifies the integral's computation, unraveling a clear pathway to integrate by parts. This technique systematically reduces complexity, transforming tough problems into manageable tasks.
When performing substitution, it is crucial to also adjust the differential. For our substitution, we derived: \( du = -x \, dx \, \text{or} \, x \, dx = -du\). This transformation allows us to rewrite the integral in terms of \(u\): \(-\int x^2 \, e^u \, du\).
Thus, the substitution method simplifies the integral's computation, unraveling a clear pathway to integrate by parts. This technique systematically reduces complexity, transforming tough problems into manageable tasks.
Definite Integrals
Definite integrals calculate the area under a curve between two bounds, helping us understand how functions accumulate values over an interval. Although our original problem dealt with an indefinite integral, knowing how definite integrals operate is key to grasping more advanced calculus concepts.
To transform an indefinite integral into a definite one, you must evaluate it at two points, usually denoted as \(a\) and \(b\). The notation \(\int_a^b f(x) \, dx\) represents the exact area under the curve \(f(x)\) from \(x = a\) to \(x = b\). This requires substituting these bounds into the integrated function and finding the difference between them.
In processes involving substitution, remember to update these limits in terms of the new variable. Always ensure the bounds reflect the transformation from \(x\) to \(u\). While this exercise didn't calculate a specific area, understanding definite integrals lays the groundwork for solving future problems where precise evaluation within intervals is necessary.
To transform an indefinite integral into a definite one, you must evaluate it at two points, usually denoted as \(a\) and \(b\). The notation \(\int_a^b f(x) \, dx\) represents the exact area under the curve \(f(x)\) from \(x = a\) to \(x = b\). This requires substituting these bounds into the integrated function and finding the difference between them.
In processes involving substitution, remember to update these limits in terms of the new variable. Always ensure the bounds reflect the transformation from \(x\) to \(u\). While this exercise didn't calculate a specific area, understanding definite integrals lays the groundwork for solving future problems where precise evaluation within intervals is necessary.
Exponential Functions
Exponential functions, characterized by the constant \(e\) being raised to a variable power, play a critical role in advanced mathematics, including calculus and differential equations. The function \(e^x\) is unique due to its derivative and integral being the same: it simplifies many mathematical models.
In our integral, the expression \(e^{-x^2/2}\) presented a classic case requiring integration by parts for solution. Here, exponential functions were managed using substitution to introduce a polynomial factor compatible with the integrand's structure. This strategic simplification unveils the essence of exponential growth or decay properties involved.
Exponential functions appear in numerous real-world applications, from population models to radioactive decay. Their natural inclination to either grow or decay aligns perfectly with advanced integration techniques, making them a common yet exciting focus in calculus.
In our integral, the expression \(e^{-x^2/2}\) presented a classic case requiring integration by parts for solution. Here, exponential functions were managed using substitution to introduce a polynomial factor compatible with the integrand's structure. This strategic simplification unveils the essence of exponential growth or decay properties involved.
Exponential functions appear in numerous real-world applications, from population models to radioactive decay. Their natural inclination to either grow or decay aligns perfectly with advanced integration techniques, making them a common yet exciting focus in calculus.
Other exercises in this chapter
Problem 60
Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x+2}{x^{2}+2} d x $$
View solution Problem 61
The integral $$ \int \ln x d x $$ can be evaluated in two ways. (a) Write \(\ln x=1 \cdot \ln x\) and use integration by parts to evaluate the integral. (b) Use
View solution Problem 63
Use an appropriate substitution to evaluate $$ \int x(x-2)^{1 / 4} d x $$
View solution Problem 64
Simplify the integrand and then use an appropriate substitution to evaluate $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{(\sin x-\cos x)^{2}} d x $$
View solution