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 |