"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!
$\int_1^2 \frac{2\pi}{x} \sqrt{1 + \frac{1}{x^4}} \, \diff x$
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.
To evaluate the definite integral $\int_1^2 \frac{2\pi}{x} \sqrt{1 + \frac{1}{x^4}} \, \mathrm{d}x$ using the fundamental theorem of calculus, you first need to find the antiderivative (or indefinite integral) of the integrand.
However, as you mentioned, finding the antiderivative of this particular function can be quite challenging and may not be easily done by hand. In situations like this, it is common to rely on computer programs or calculators to compute the antiderivative.
One suggestion is to use mathematics software like Wolfram Alpha or Mathematica. These programs have built-in algorithms that can handle complicated integrals and provide the result symbolically or numerically. You can input your integral into the program as:
```
integrate (2π/x) * sqrt(1 + 1/x^4) dx from 1 to 2
```
This will give you the exact value of the definite integral.
Keep in mind that while using computer programs is a practical approach to finding the antiderivative, it is important to understand the concept of antiderivatives and know how to evaluate them manually. In many cases, the antiderivative may not have a closed-form expression, and numerical methods or approximation techniques might be necessary.
Once you have the antiderivative, you can evaluate the definite integral using the fundamental theorem of calculus by simply subtracting the antiderivative evaluated at the upper limit (2) from the antiderivative evaluated at the lower limit (1).
I hope this helps!