What is the derivative of ln(4x) ?sashab said:Homework Statement
Use integration by parts to evaluate the integral.
∫5x ln(4x)dx
Homework Equations
∫udv = uv - ∫vdu
The Attempt at a Solution
So here's my solution:
But the computer is telling me I'm wrong :( We haven't learned how to integrate lnx yet, so the only choice I have is to make u = ln(4x) (even our textbook does this). Any help would be really appreciated! Thanks :)
SammyS said:
Integration by Parts is a method used to solve integrals that involve products of functions. It uses the formula ∫u dv = uv - ∫v du, where u and v are chosen functions and du and dv are their respective differentials.
To solve Integration by Parts, you need to follow these steps:
The purpose of using Integration by Parts is to solve integrals that cannot be solved by other integration techniques, such as substitution or u-substitution. It allows us to break down a complex integral into simpler parts and solve it using the formula ∫u dv = uv - ∫v du.
No, Integration by Parts can only be used for integrals that involve products of functions. It is not applicable for integrals that involve quotients or nested functions.
To use Integration by Parts to solve 5x ln(4x)dx, we need to choose u and dv based on the LIATE rule. In this case, we can choose u = ln(4x) and dv = 5x dx. Then, we calculate du = (1/x)dx and v = (5/2)x^2. Substituting these values into the formula ∫u dv = uv - ∫v du, we get the simplified integral 5/2 x^2 ln(4x) - 5/2 x^2 dx. Finally, we solve for x using basic integration techniques to get the final answer.