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Magazine Chapter 7: Problem 17
Find the integral. $$ \int x \cdot 5^{-x^{2}} d x $$
Short Answer
Expert verified
The integral is \(-\frac{5^{-x^2}}{2 \ln 5} + C\.\)
Step by step solution
01
Recognize the Integral Type
The given integral is of the form \( \int x \cdot a^{-x^2} \, dx \), where \( a \) is a constant (5 in this case). This suggests that it can be solved using substitution if recognized as a Gaussian integral.
02
Choose a Substitution
To simplify the integral, use the substitution \( u = -x^2 \). Consequently, differentiate this substitution: \( du = -2x \, dx \). Rearrange to find \( x \, dx \): \( x \, dx = -\frac{1}{2} du \).
03
Rewrite the Integral with Substitution
Substitute \( u \) and \( du \) into the integral: \[ \int x \cdot 5^{-x^2} \, dx = \int 5^u \cdot \left( -\frac{1}{2} du \right) = -\frac{1}{2} \int 5^u \, du. \]
04
Integrate with Respect to u
The integral \( \int 5^u \, du \) results in \( \frac{5^u}{\ln 5} + C \), where \( C \) is the constant of integration. Thus, \[ -\frac{1}{2} \int 5^u \, du = -\frac{1}{2} \left( \frac{5^u}{\ln 5} \right) + C = -\frac{5^u}{2 \ln 5} + C. \]
05
Reverse the Substitution
Recall the substitution \( u = -x^2 \). Substitute back to get the integral in terms of \( x \): \[ -\frac{5^{-x^2}}{2 \ln 5} + C. \]
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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 in calculus that is frequently employed to simplify complex integrals. The idea is to transform a challenging integral into a simpler form that is easier to evaluate. By choosing an appropriate substitution, we can rewrite both the integrand and the differential.
- Start by identifying a portion of the integral that, when substituted, simplifies the problem.
- For the integral \( \int x \cdot 5^{-x^{2}} \, dx \), substitute \( u = -x^{2} \). This choice simplifies the exponent.
- Differentiate \( u \) with respect to \( x \): \( du = -2x \, dx \), and rearrange it to find \( x \, dx = -\frac{1}{2} du \).
- Insert these into the integral to rewrite it in terms of \( u \): \( \int 5^u \cdot \left( -\frac{1}{2} du \right) \).
Gaussian Integral
Gaussian integrals are a special class of integrals that involve an exponential function with a quadratic expression in the exponent, commonly seen in the form \( e^{-x^2} \). These integrals are named after the mathematician Carl Friedrich Gauss.
- In the exercise, the integral mimics a Gaussian form \( a^{-x^2} \), where \( a = 5 \).
- Although not an exact Gaussian integral due to the base 5, the substitution method effectively deals with a similar form.
- The solution involves changing variables to transform the integral into a more typical exponential form \( 5^u \).
Constant of Integration
The constant of integration, often denoted as \( C \), is an essential component when dealing with indefinite integrals. It accounts for the family of all possible antiderivatives of a function.
- During integration, especially with indefinite integrals, the process reverses differentiation, which eliminates any constant term present in the original function.
- By introducing \( C \), we acknowledge that any constant could have been a part of the original function before differentiation.
- In the exercise, after integrating with respect to \( u \), the expression includes \( C \): \( -\frac{5^u}{2 \ln 5} + C \).
