I cannot for the life of me figure this out. Please help me.
How do I integrate the function f(x) = 0.1 * e ^ (-0.2 * |x|) from neg. Infinity to pos. Infinity?
I seem to only be able to get 0, but the answer is 1. I think it is the |x| that is throwing me off. Please Hlep!
Integrate it in two parts: minus inf to zero, and zero to inf. On the negative side, change the sign of the exponent, just as the abs value sign did.
Integrating the function f(x) = 0.1 * e^(-0.2 * |x|) can indeed be a bit tricky, but fear not! Clown Bot is here to help with a touch of humor!
Let's break it down into two parts - the negative side and the positive side.
On the negative side (from negative infinity to zero), you are correct in changing the sign of the exponent due to the absolute value sign. So, the integral becomes:
∫[negative infinity to 0] 0.1 * e^(0.2 * x) dx
Now, let's introduce Clown Bot's friend, the substitution! Let u = 0.2 * x, which means du = 0.2 * dx. We can rewrite the integral as:
∫[negative infinity to 0] 0.1 * e^u * (0.2 du / 0.2)
Simplifying a bit, we get:
∫[negative infinity to 0] 0.1 * e^u du
Now, the integral of e^u is just e^u, so we have:
0.1 * e^u evaluated from negative infinity to 0
Plugging in the limits of integration, we get:
0.1 * e^0 - 0.1 * e^(-infinity)
But hey, Clown Bot is all about fun, and e^(-infinity) is basically 0, making this part of the integral:
0.1 * 1 - 0.1 * 0 = 0.1
Now, onto the positive side (from zero to positive infinity). The absolute value sign disappears here since x is positive, so we have:
∫[0 to positive infinity] 0.1 * e^(-0.2 * x) dx
This is more straightforward, and we can integrate it directly. The integral of e^(-0.2 * x) is -5 * e^(-0.2 * x), so we have:
-0.1 * e^(-0.2 * x) evaluated from 0 to positive infinity
Now, as x goes to positive infinity, e^(-0.2 * x) goes to 0, which makes this part of the integral:
-0.1 * 0 - (-0.1 * 1) = 0.1
Now, all we have to do is add the two parts together:
0.1 + 0.1 = 0.2
Oh no, Clown Bot made a mistake! The answer is not 1, but 0.2. Apologies for the mix-up!
There you have it! The correct answer to the integral of f(x) = 0.1 * e^(-0.2 * |x|) from negative infinity to positive infinity is 0.2. I hope you found this explanation helpful, even though Clown Bot couldn't resist adding a little twist of humor! Keep up the great work, and never stop exploring the magical world of mathematics!
To integrate the function f(x) = 0.1 * e^(-0.2 * |x|) from negative infinity to positive infinity, you are correct that you need to consider two separate integrals: one from negative infinity to zero and another from zero to positive infinity.
First, let's integrate from negative infinity to zero. In this interval, the absolute value of x is simply -x. So the function becomes f(x) = 0.1 * e^(-0.2 * -x) = 0.1 * e^(0.2x).
Now, to integrate this function from negative infinity to zero, we can use the standard integral of e^x. The integral of e^x is e^x, so we have:
∫[negative infinity to 0] 0.1 * e^(0.2x) dx = 0.1 * [e^(0.2x)/0.2] [negative infinity to 0]
Applying the limits, we get:
[0.1 * (e^(0.2 * 0)/0.2)] - [0.1 * (e^(0.2 * negative infinity)/0.2)]
Simplifying further, we have:
[0.1 * (1/0.2)] - [0.1 * (e^(-infinity)/0.2)]
= 0.5 - 0
= 0.5
Now, let's integrate from zero to positive infinity. In this interval, the absolute value of x is simply x. So the function remains f(x) = 0.1 * e^(-0.2 * x).
To integrate this function from zero to positive infinity, we can again use the standard integral of e^x. The integral of e^x is e^x, so we have:
∫[0 to positive infinity] 0.1 * e^(-0.2x) dx = -0.1 * [e^(-0.2x)/0.2] [0 to positive infinity]
Applying the limits, we get:
-0.1 * [(e^(-0.2 * positive infinity))/0.2] - (-0.1 * (e^(-0.2 * 0))/0.2)]
We know that e raised to any negative power approaches zero, so:
-0.1 * [0/0.2] - (-0.1 * (e^0)/0.2)]
= -0.1 * 0 - (-0.1 * 1/0.2)
= 0 - (-0.05)
= 0.05
Finally, to find the total integral from negative infinity to positive infinity, we add the results from the two separate integrals:
0.5 + 0.05 = 0.55
So, the total integral of f(x) = 0.1 * e^(-0.2 * |x|) from negative infinity to positive infinity is 0.55, not 1 as you mentioned.
To integrate the function f(x) = 0.1 * e ^ (-0.2 * |x|) over the range from negative infinity to positive infinity, we can break it into two separate integrals: one from negative infinity to zero and another from zero to positive infinity.
Let's start with the first integral. For x < 0, the absolute value sign becomes -x. So, we can rewrite the integrand as f(x) = 0.1 * e ^ (-0.2 * (-x)) = 0.1 * e ^(0.2x).
The integral from negative infinity to zero of f(x) can now be written as ∫[negative infinity to zero] 0.1 * e ^(0.2x) dx.
Evaluating this integral, we can use the standard calculus techniques. We need to find the integral of e^(0.2x). The integral of e^(kx)dx is (1/k)e^(kx) + C, where k is a constant.
Using this formula, the integral of 0.1 * e ^(0.2x) dx becomes (0.1/0.2) * e ^(0.2x) from negative infinity to zero.
For the lower limit, as x approaches negative infinity, e^(0.2x) approaches zero. So, the lower limit term becomes (0.1/0.2) * e ^(0) = 0.1/0.2 = 0.5.
Now for the upper limit, as x approaches zero, e^(0.2x) becomes 1. So, the upper limit term becomes (0.1/0.2) * 1 = 0.1/0.2 = 0.5.
Therefore, the first integral from negative infinity to zero evaluates to 0.5 - 0.5 = 0.
Now let's move on to the second integral from zero to positive infinity. The function f(x) remains the same, so the integrand is still 0.1 * e ^(0.2x).
Using the same technique as before, we evaluate the integral from zero to infinity of f(x) as ∫[zero to infinity] 0.1 * e ^(0.2x) dx.
Now, for the lower limit, as x approaches zero, e^(0.2x) becomes 1. So, the lower limit term becomes (0.1/0.2) * 1 = 0.1/0.2 = 0.5.
For the upper limit, as x approaches positive infinity, e^(0.2x) approaches infinity.
Therefore, the second integral from zero to positive infinity evaluates to infinity - 0.5 = infinity.
Now, let's add the results of the two integrals together: 0 + infinity.
When evaluating a sum of zero and infinity, the result is undefined, not equal to 1 as you mentioned in your question.
It seems there might be a mistake or misunderstanding in the problem statement or the expected answer you mentioned. Double-check the question or consult with the appropriate sources to clarify any discrepancies.