What's the integral of e^x*ln(x)?

The integral of e^x*ln(x) is e^x*(ln(x)-1) + C.

To solve this integral, we can use integration by parts. Let u = ln(x) and dv = e^x dx. Then du = 1/x dx and v = e^x. Using the formula for integration by parts, we have:

∫ e^x*ln(x) dx = e^x*ln(x) - ∫ e^x*(1/x) dx

The second integral can be solved using integration by substitution. Let u = x, then du = dx and the integral becomes:

∫ e^x*(1/x) dx = ∫ e^u du = e^u + C = e^x + C

Substituting this back into the original equation, we have:

∫ e^x*ln(x) dx = e^x*ln(x) - e^x + C

Simplifying this expression, we get:

∫ e^x*ln(x) dx = e^x*(ln(x)-1) + C

Therefore, the integral of e^x*ln(x) is e^x*(ln(x)-1) + C.

For a deeper understanding of the techniques used in this integration, explore our notes on Techniques of Integration.

A-Level Maths Tutor Summary: In summary, to find the integral of e^xln(x), we apply integration by parts. This technique involves choosing parts of the function to differentiate and integrate separately, then combining them. The process simplifies to e^x(ln(x)-1) + C. This method breaks down complex integrals into easier steps, allowing us to solve them step by step.

To learn more about different integration methods, including those suitable for similar problems, you might find our discussion on Definite and Indefinite Integrals helpful. Additionally, further details on integration rules can be accessed through our detailed notes on Integration Rules.