Solving Reverse Integrals: Find f(x) to Solve 1-0.1^n

In summary, in order to find a function f(x) such that the integral from negative infinity to (100+10n) of f(x) is equal to 1-0.1^n for n=1,2,3,4,5,6...∞, you can replace (100+10n) with a real variable (y) and take the derivative of both sides to get an expression for f(y). This expression will be an exponential function, specifically f(y) = exp((y-100)ln(0.1)/10). Keep in mind the integral limits of negative infinity to y when solving for f(y).
  • #1
Big-Daddy
343
1
I need to find a function f(x) such that

[tex]\int_{-\infty}^{100+10n} (f(x)) dx = 1-0.1^n[/tex]

for n=1,2,3,4,5,6...∞. How would I go about this? It must be exponential in some way I'm guessing?

This is not a homework problem. I don't just want the answer. I want guidance on this type of problem and function, but please from someone with an idea of how to answer this particular case too ...
 
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  • #2
Replace (100+10n) by a real variable (y). Take the derivative of both sides, this will give you an expression for f(y) if it exists. It will be an exponential.
 
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  • #3
mathman said:
Replace (100+10n) by a real variable (y). Take the derivative of both sides, this will give you an expression for f(y) if it exists. It will be an exponential.

So y=100+10n, n = 1/10(y-100) = (1/10)y-10.

Now what? I need to bear the integral limits (-∞ to y) in mind...
 
  • #4
0.1n = 0.1(y-100)/10 = exp((y-100)ln(0.1)/10). Now take the derivatives of both sides to get f(y) = .
 
  • #5


I understand the importance of finding a solution to a problem rather than just the answer. In this case, we are trying to find a function f(x) that would satisfy the given integral for different values of n. This type of problem falls under the category of "reverse integrals" and can be challenging to solve.

One approach to solving this problem would be to use the Fundamental Theorem of Calculus, which states that the derivative of the integral of a function is the original function. In other words, if we can find a function g(n) such that g'(n) = f(x), then we can solve the integral by simply evaluating g(n) at the given limits of integration.

In this case, we can start by rewriting the given integral as:

\int_{-\infty}^{100+10n} (f(x)) dx = \int_{-\infty}^{100+10n} (g'(n)) dx

We can then use the chain rule to find the derivative of g(n) with respect to n. This would give us:

g'(n) = f(x) = \frac{d}{dn} \int_{-\infty}^{100+10n} (g'(n)) dx = 10g'(n)

Now, we can see that g'(n) is a multiple of itself, which suggests an exponential function. We can rewrite g'(n) as:

g'(n) = Ce^{10n}

where C is a constant. We can then integrate g'(n) to find g(n):

g(n) = Ce^{10n}/10 + K

where K is another constant.

Finally, we can substitute g(n) back into the original integral to get:

\int_{-\infty}^{100+10n} (f(x)) dx = \int_{-\infty}^{100+10n} (g'(n)) dx = \int_{-\infty}^{100+10n} (Ce^{10n}) dx = Ce^{10n}(100+10n) + K

Since we want this integral to equal 1-0.1^n, we can set K = 0 and C = 1/101. This would give us the final solution of:

f(x) = \frac{1}{101}e^{10n}

Therefore,
 

1. How do I solve a reverse integral to find f(x)?

To solve a reverse integral, you need to follow these steps:
1. Rewrite the integral in its differential form.
2. Integrate the function using the power rule.
3. Add a constant of integration.
4. Solve for the constant using the given conditions.
5. Rewrite the solution in terms of f(x).

2. What is the formula for solving a reverse integral?

The formula for solving a reverse integral is: f(x) = ∫(1-0.1^n)dx = x - 0.1^n+1/n+1 + C

3. Can I use any integration technique to solve a reverse integral?

Yes, you can use any integration technique such as substitution, integration by parts, or partial fractions to solve a reverse integral. However, it is important to choose the appropriate technique based on the given function.

4. How do I find the constant of integration when solving a reverse integral?

The constant of integration can be found by using the given conditions. For example, if the value of f(x) is given at a certain point, you can substitute the value into the solution and solve for the constant.

5. Can I use a calculator to solve a reverse integral?

Yes, you can use a calculator to solve a reverse integral. However, it is important to note that the calculator may not provide the constant of integration, so you will need to manually solve for it using the given conditions.

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