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reidwilson
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How do I calculate the intergral of e^(-|x|) ("e to the minus absolute value of x") without a calculator?
Calculate Integral of e^(-|x|) Without a Calculator
- Thread starter reidwilson
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In summary, the integral of e^(-|x|) is equal to 2e^(-|x|) + C, and it can be calculated without a calculator using the substitution method. It can also be solved using integration by parts, resulting in the integral -xe^(-|x|) + e^(-|x|) + C. The integral is an even function and is commonly used in various applications, such as modeling decay rates and solving differential equations.
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tiny-tim
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welcome to pf!
hi reidwilson! welcome to pf!
(try using the X2 icon just above the Reply box)
same way as ∫ e-x dx or ∫ ex dx,
hi reidwilson! welcome to pf!
(try using the X2 icon just above the Reply box
)reidwilson said:
same way as ∫ e-x dx or ∫ ex dx,
except you need to do the cases of x < 0 and x > 0 separately
FAQ: Calculate Integral of e^(-|x|) Without a Calculator
1. What is the integral of e^(-|x|)?
The integral of e^(-|x|) is equal to 2e^(-|x|) + C, where C is a constant of integration.
2. How do you calculate the integral of e^(-|x|) without a calculator?
To calculate the integral of e^(-|x|) without a calculator, you can use the substitution method. Let u = -|x|, then du = -dx. This will simplify the integral to -e^u + C, which can be easily evaluated.
3. Can the integral of e^(-|x|) be solved using integration by parts?
Yes, the integral of e^(-|x|) can also be solved using integration by parts. Let u = e^(-|x|), then du = -e^(-|x|)dx and dv = dx, which can be integrated to v = x. This will give the integral as -xe^(-|x|) + e^(-|x|) + C.
4. Is the integral of e^(-|x|) an even or odd function?
The integral of e^(-|x|) is an even function, as the absolute value of x is eliminated during the integration process. This means that the integral is symmetrical about the y-axis.
5. Why is the integral of e^(-|x|) used in many applications?
The integral of e^(-|x|) is used in many applications because it models the decay rate of certain processes, such as radioactive decay. It is also a common solution for differential equations involving exponential functions. Additionally, it is often used as a basis for other integrals and can be used to solve more complex integrals through substitution or integration by parts.