The formula for integration by parts is ∫udv = uv - ∫vdu, where u is the first function and dv is the derivative of the second function.
When using integration by parts, it is important to choose u and dv in a way that will simplify the integral or make it easier to solve. As a general rule, u should be the function that becomes simpler when differentiated and dv should be the function that becomes easier to integrate.
To integrate ln(x)/x^4 using integration by parts, we can choose u = ln(x) and dv = 1/x^4. Then, we can use the formula ∫udv = uv - ∫vdu to solve the integral. After integrating by parts, we will have a new integral to solve, which can be simplified by using substitution or other integration techniques.
Integration by parts allows us to solve integrals that cannot be solved using other integration techniques, such as substitution or the power rule. It also allows us to simplify integrals and make them easier to solve.
One common mistake to avoid when using integration by parts is choosing u and dv in the wrong order, which can lead to a more complicated integral. It is also important to be careful with signs and make sure to simplify the integral after using the integration by parts formula.