Is the Integral of (sinx)/(1+cos^2(x)) convergent or divergent?

• tbone413
In summary, the problem is asking whether the given integral is convergent or divergent, and if convergent, to solve for the integral from 0 to infinity. The solution involves finding the antiderivative, using the substitution u=cos(x), and evaluating it between 0 and L before taking the limit as L approaches infinity. The answer is pi/4, as given in the book. It is determined to be convergent through various convergence tests such as the ratio test and p test.
tbone413

Homework Statement

State whether the problem is convergent or divergent and if its convergent solve the integral. Integral from 0 to inf of (sinx)/(1+cos^2(x))dx

Homework Equations

there isn't really an equation for this, I don't think.

The Attempt at a Solution

Im not really sure how to even start this problem. I know it converges, and I know the answer is pi/4 (the book has the answer) but I am stuck on how to figure out that it A) converges and B) how to solve it.

*I know there are several tests you can use to test for convergence (ratio test, p test, etc.) but I am not sure which one applies here.

It doesn't converge. Find the antiderivative, use u=cos(x). Evaluate it between 0 and L and then let L->infinity. Does it approach a limit?

What is "convergence of integral"?

The convergence of an integral refers to whether the value of a definite integral is finite or infinite. It is determined by the behavior of the integrand (the function being integrated) as well as the limits of integration.

How is the convergence of an integral tested?

The most common methods for testing the convergence of an integral include the comparison test, the limit comparison test, and the integral test. These tests involve comparing the given integral to a known convergent or divergent integral, or evaluating the limit of the integrand.

What is the significance of convergence of an integral?

The convergence of an integral is important because it determines whether the solution to a problem is valid or not. A convergent integral indicates that the solution exists and can be calculated, while a divergent integral implies that the solution does not exist or is infinite.

What is the difference between absolute and conditional convergence of an integral?

Absolute convergence refers to the convergence of an integral regardless of the order in which the terms are added, while conditional convergence only holds when the terms are added in a specific order. In other words, an absolutely convergent integral will have the same value regardless of how the limits of integration are rearranged, while a conditionally convergent integral may have different values depending on the order of integration.

What happens if an integral does not converge?

If an integral does not converge, it is said to be divergent. This means that the value of the integral is either infinite or does not exist. In such cases, the problem being solved may not have a valid solution, or alternative methods may need to be used to find a solution.

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