Posted by **Parker** on Tuesday, July 3, 2012 at 6:03pm.

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

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!

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

- Calculus -
**Count Iblis**, Tuesday, July 3, 2012 at 7:07pm
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.

## Answer this Question

## Related Questions

- calculus - using the fundamental theorem of calculus what is the derivative of ...
- Calculus I - Section The fundamental Theorem of Calculus: Use Part I of the ...
- Calculus Fundamental Theorem - Evaluate the definite integral. function: x+13 ...
- Calculus - definite Integrals (using fundamental Theorem) Evaluate from -1 to 2(...
- Calculus - assume the rate at which the world's oil is being consumed is r = 27 ...
- Calculus Fundamental Theorem - Evaluate the definite integral. function: (t+8)(t...
- calculus (please with steps and explanations) - consider the function f that is ...
- Calculus (urgent help) - consider the function f that is continuous on the ...
- Calculus - for the function F(t)=ln(t^2), let f(t)=F'(t). Write the integral ...
- calculus - consider the function f that is continuous on the interval [-5,5] and...