A Difficult Calculus Problem

Let 

and let g(x) be an antiderivative of f(x). Then if

        g(5)  =  7

find

        g(1)

Solution

Since g(x) is an antiderivative of f(x), we have 

        g '(x)  =  f(x)

or

None of the regular techniques of integration will work on this integral.  Even the computer cannot solve this explicitly.  Instead of integrating, we let

        h(x)  =  g(x) - 7

Then h(x) is also an antiderivative of f(x) and 

        h(5)  =  0

We can write

Notice that when we plug in 5 for x, we get 0 as required, since the upper and lower limits are equal.  Now use a calculator to easily find

finally since 

        h(x)  =  g(x) - 7

it follows that 

        g(x)  =  h(x) + 7

and that 

        g(1)  =  h(1) + 7  =  -10.88222 + 7  =  -3.88222

Back to the Second Fundamental Theorem of Calculus