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Homework Statement
For which value the definite integral (0 to infinity) e^ax cos(x) dx converges? Calculate for the value of a.
Mathieu
In mathematics, a series or integral is said to converge if its terms or integrands approach a definite value, usually referred to as the limit. In simpler terms, it means that the series or integral has a well-defined and finite value.
The value of a is directly related to the convergence of the integral. If a is equal to 0, the integral will always converge. However, if a is greater than 0, the integral will only converge for certain values of a. If a is less than 0, the integral will not converge for any value of a.
Yes, for example, if a = -1, the integral would converge to e^(-x)sin(x). Other values of a that would make the integral converge include a = -2, a = -3, and so on.
Yes, there is a general rule for determining the values of a that would make the integral converge. If the value of a is between -1 and 1, the integral will converge. If a is equal to -1 or 1, the integral will converge conditionally. If a is less than -1 or greater than 1, the integral will not converge.
If the value of a is not within the range for convergence, the integral will not converge. This means that the integral will not have a well-defined and finite value. It may either diverge to infinity or oscillate between different values.