Don't forget the dx.Neil21 said:Ok, so I complete the square and got [tex] \int \frac{1}{((x^{2}+\frac{1}{2})^{2}+\frac{3}{4})^{\frac{1}{2}}} [/tex] . I am still not seeing a way find an antiderivative though.
What I first saw was rewriting the expression as $${\frac{1}{(x^4+x^2+1)^{1/2}}}={\frac{1}{((x^2+({\frac{1}{2}))^2}+(\frac{\sqrt{3}}{2})^2)^{1/2}}}$$.Mark44 said:Don't forget the dx.
$$\int \frac{dx}{((x^{2}+\frac{1}{2})^{2}+\frac{3}{4})^{\frac{1}{2}}}$$
I haven't worked this through, yet, but here's what I would do, FWIW.
Let u = x2 + 1/2
So ##x = \sqrt{u - 1/2}##
Then du = 2xdx
So ##dx = \frac{du}{2x} = \frac{du}{2\sqrt{u - 1/2}}##
After making the substitution and getting everything in terms of u and du, I would try for a trig substitution. I think that might work.
Ooops. Missed that. Sorry guys.Mark44 said:@chet, I think he already established that in the opening post.
The convergence of an integral refers to whether or not the value of the integral exists or is finite. In other words, it is a measure of whether the function being integrated approaches a finite limit as the interval of integration becomes smaller.
The convergence of an integral is determined by evaluating the integral using various methods such as the comparison test, the limit comparison test, or the integral test. These methods help determine if the integral converges or diverges.
Absolute convergence of an integral means that the integral converges regardless of the order of integration. Conditional convergence, on the other hand, means that the integral only converges when integrated in a specific order.
Yes, an integral can converge to a negative value. The convergence of an integral is determined by the limit of the function being integrated, so it is possible for the limit to be negative, resulting in a negative value for the integral.
The convergence of an integral has many applications in physics, engineering, and economics. For example, it is used to calculate the area under a curve, which is important in determining work done, finding the center of mass, and calculating probabilities in economics.