SOLUTION: Please help me solve this equation: Consider the function {{{ f(x) = e^(-x)/(1-e^-5) }}} {{{ 0<=x<=5 }}} a) Verify that {{{ int( f(x), dx, 0, 5 ) =1 }}} b) Find {{{ int ( x f(

Question 355597: Please help me solve this equation:
Consider the function
a) Verify that
b) Find
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

For both parts it helps to recognize that the denominator of f(x) is a constant. So we can rewrite the function as:

a) Verify that

We can factor out the constant from the integrand:

Now the integral is easy to find:
evaluated from 0 to 5:

since

which simplifies to 1 (since .

b) Find

Again we can factor out the constant:

This integral is a job for Integration by Parts. If u = x and dv = . This makes v = and du = dx. Substituting u and dv we get:

(Technically the boundary values, 0 to 5, change, too. But since I am going to substitute back in for u and v I am not going to spend the time to make this adjustment. You, however, should not skip this step.)
By the rule for Integration by Parts, , we get:

The remaining integral is easy:

Evaluating this from 0 to 5:

If we multiply the numerator and denominator by we get:

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