# How do you test for convergence: #int (((sin^2)x)/(1+x^2)) dx#?

The fact that this last integral converges can be seen by direct calculation:

By signing up, you agree to our Terms of Service and Privacy Policy

To test for the convergence of the integral [ \int \frac{\sin^2(x)}{1+x^2} , dx ]

We can use the Comparison Test or the Limit Comparison Test. Here, we will use the Comparison Test.

**Comparison Test**:

Since ( |\sin(x)| \leq 1 ) for all ( x ), we have: [ \frac{\sin^2(x)}{1+x^2} \leq \frac{1}{1+x^2} ]

Now, consider the integral: [ \int \frac{1}{1+x^2} , dx ]

This integral is a standard integral which converges. Therefore, by the Comparison Test, if we can show that [ \int \frac{1}{1+x^2} , dx ] converges, then [ \int \frac{\sin^2(x)}{1+x^2} , dx ] also converges.

The integral [ \int \frac{1}{1+x^2} , dx ] can be evaluated as: [ \int \frac{1}{1+x^2} , dx = \arctan(x) + C ]

This integral converges as ( x ) approaches infinity.

Thus, by the Comparison Test, [ \int \frac{\sin^2(x)}{1+x^2} , dx ] also converges.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Using the integral test, how do you show whether #n/(n^2+1)# diverges or converges?
- What is the root test?
- How do you determine if the improper integral converges or diverges #int 5x^(2)e^(-x^(3))# from 1 to infinity?
- Evaluate the integral or show that it is divergent?
- How would I use the Comparison Test in calculus to solve the integral #(cos(4x) +3) / (3x^3 + 1)# from 1 to infinity?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7