How to integrate (e^x)/x?

The integral of (e^x)/x cannot be expressed in terms of elementary functions.

Unfortunately, the integral of (e^x)/x cannot be expressed in terms of elementary functions. This means that there is no simple formula or method to find the exact value of the integral. However, it is possible to approximate the value of the integral using numerical methods such as Simpson's rule or the trapezoidal rule.

One way to approach the integral is to use integration by parts. Let u = 1/x and dv = e^x dx. Then du/dx = -1/x^2 and v = e^x. Using the formula for integration by parts, we have:

∫ (e^x)/x dx = e^x/x - ∫ e^x (-1/x^2) dx
= e^x/x + ∫ e^x/x^2 dx

This new integral can be solved using another application of integration by parts. Let u = 1/x^2 and dv = e^x dx. Then du/dx = 2/x^3 and v = e^x. Using the formula for integration by parts again, we have:

∫ e^x/x^2 dx = -e^x/x - 2∫ e^x/x^3 dx

This process can be repeated indefinitely, resulting in an infinite series of integrals. However, this series does not converge to a finite value, so it cannot be used to find the exact value of the integral.

For a deeper understanding of different techniques that might be employed in integrating functions similar to (e^x)/x, you may find it beneficial to explore various integration techniques.

A-Level Maths Tutor Summary: The integral of (e^x)/x can't be neatly solved using common maths methods, which means there isn't a straightforward answer. While techniques like integration by parts offer a way to tackle it, they lead to an endless series of more complex integrals without a clear end result. If you're looking for a guide on integration rules or require an overview of indefinite integration, these resources can be incredibly helpful. Essentially, it's a complex problem that requires advanced methods to estimate its value.