In using the technique of integration by parts, you must carefully choose which expression is $u$. For the following problem, use the guidelines in this section to choose $u$. Do not evaluate the integral.
$\int x^5 \ln (4 x) d x$
Answer (2)
To choose u for the integral ∫ x 5 ln ( 4 x ) d x , we apply the LIATE rule which prioritizes logarithmic functions over algebraic ones. Therefore, we select u = ln ( 4 x ) . This choice facilitates a smoother integration process.
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Identify the integral: ∫ x 5 ln ( 4 x ) d x .
Apply the LIATE rule to prioritize functions for 'u': Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.
Recognize ln ( 4 x ) as logarithmic and x 5 as algebraic.
Choose u = ln ( 4 x ) based on LIATE.
The choice of u is ln ( 4 x ) .
Explanation
Problem Analysis We are given the integral ∫ x 5 ln ( 4 x ) d x and asked to choose the expression for u when using integration by parts.
Integration by Parts Formula and LIATE Rule The integration by parts formula is ∫ u d v = uv − ∫ v d u . The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) helps us choose u .
Applying LIATE Rule In the given integral, we have a logarithmic function ln ( 4 x ) and an algebraic function x 5 . According to the LIATE rule, we should choose the logarithmic function as u before the algebraic function.
Choosing u and dv Therefore, we choose u = ln ( 4 x ) and d v = x 5 d x .
Examples
Integration by parts is a technique used to integrate the product of two functions. For example, in economics, you might use integration by parts to calculate the present value of a continuous income stream, where one function represents the income rate and the other represents the discount factor. Choosing the correct 'u' and 'dv' is crucial for simplifying the integral and making it easier to solve. This technique is also used in physics to solve problems involving work and energy.
