"Leave the answer as a definite integral, but indicate how it could by evaluated by using the fundamental theorem of calculus."

Question

I solved the problem to a definite integral. Proceeding via the fundamental theorem, would involve finding the indefinite integral (or antiderivative). I can use a computer program to do this, but I don't see a reasonable way to do this by hand. Any suggestions? thanks!

$\int_1^2 \frac{2\pi}{x} \sqrt{1 + \frac{1}{x^4}} \, \diff x$

Answer 1

Substituting x = sqrt(t) leads to an integral of the form:

Integral of dt/t sqrt(1+t^2)

If you then put t = sinh(u), this becomes:

Integral of cosh^2(u)/sinh(u)du =

Integral of [1/sinh(u) + sinh(u)] du

Then the integral of 1/sinh(u) be evaluated by putting u = Log(v):

du/[exp(u) - exp(u)] =

dv/[v (v - 1/v)] = dv/(v^2 - 1)

which is easily integrated.