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bolbteppa
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On the bottom of page 24 & top of page 25 of this pdf an integral is beautifully computed by breaking it up into an infinite series. Is there any reference where I could get practice in working integrals like these?
"Integrals via Infinite Series" is a mathematical technique used to approximate the value of a definite integral by representing it as an infinite series. This technique is based on the fundamental theorem of calculus, which states that the integral of a function can be calculated by finding its antiderivative. By using infinite series, we can approximate the antiderivative and thus, the value of the integral.
This technique is typically used when the function being integrated is difficult to integrate by traditional methods, such as integration by parts or substitution. It is also useful for functions that do not have a closed-form antiderivative. In these cases, using an infinite series to approximate the integral can be more efficient and accurate.
Some common examples include integrals involving trigonometric functions, logarithmic functions, and exponential functions. For example, the integral of sin(x) can be approximated using an infinite series, as can the integral of ln(x) and e^x.
The process involves representing the function being integrated as an infinite series, then evaluating the series to find an approximation for the antiderivative. This approximation can then be used to find an approximation for the value of the integral. The accuracy of the approximation can be improved by using more terms in the series.
One limitation is that this technique can only be used for definite integrals, not indefinite integrals. Additionally, the convergence of the infinite series must be carefully considered in order to ensure an accurate approximation. In some cases, the series may not converge or may converge very slowly, making this technique less practical. It is also important to note that this method is an approximation and may not give an exact solution.