Find the exact value of $\int_1^3 x^2 \ln 3 x d x$. Give your answer in the form $a \ln b+c$, where $a$ and $c$ are rational and $b$ is an integer.
Answer (1)
Apply integration by parts with u = ln 3 x and d v = x 2 d x , leading to d u = x 1 d x and v = 3 x 3 .
Use the integration by parts formula: ∫ u d v = uv − ∫ v d u to get ∫ 1 3 x 2 ln 3 x d x = [ 3 x 3 ln 3 x ] 1 3 − ∫ 1 3 3 x 2 d x .
Evaluate the integral and the term separately: ∫ 1 3 3 x 2 d x = 9 26 and [ 3 x 3 ln 3 x ] 1 3 = 3 53 ln 3 .
Combine the results to express the final answer in the required form: 3 53 ln 3 − 9 26 .
Explanation
Problem Analysis We are asked to find the exact value of the definite integral ∫ 1 3 x 2 ln 3 x d x and express the result in the form a ln b + c , where a and c are rational and b is an integer.
Integration by Parts To solve this integral, we will use integration by parts. The formula for integration by parts is ∫ u d v = uv − ∫ v d u . Let's choose u = ln 3 x and d v = x 2 d x . Then, we find d u and v .
Finding du and v If u = ln 3 x , then d u = 3 x 1 ⋅ 3 d x = x 1 d x . If d v = x 2 d x , then v = ∫ x 2 d x = 3 x 3 .
Applying Integration by Parts Formula Now we apply the integration by parts formula: ∫ 1 3 x 2 ln 3 x d x = [ 3 x 3 ln 3 x ] 1 3 − ∫ 1 3 3 x 3 ⋅ x 1 d x = [ 3 x 3 ln 3 x ] 1 3 − ∫ 1 3 3 x 2 d x
Evaluating the Integral Next, we evaluate the integral ∫ 1 3 3 x 2 d x :
∫ 1 3 3 x 2 d x = 3 1 ∫ 1 3 x 2 d x = 3 1 [ 3 x 3 ] 1 3 = 3 1 ( 3 3 3 − 3 1 3 ) = 3 1 ( 3 27 − 3 1 ) = 3 1 ( 3 26 ) = 9 26
Evaluating the Term Now we evaluate the term [ 3 x 3 ln 3 x ] 1 3 :
[ 3 x 3 ln 3 x ] 1 3 = 3 3 3 ln ( 3 ⋅ 3 ) − 3 1 3 ln ( 3 ⋅ 1 ) = 9 ln 9 − 3 1 ln 3 = 9 ln 3 2 − 3 1 ln 3 = 18 ln 3 − 3 1 ln 3 = 3 54 ln 3 − 3 1 ln 3 = 3 53 ln 3
Combining the Results Combining the results, we have: ∫ 1 3 x 2 ln 3 x d x = 3 53 ln 3 − 9 26
Final Answer Thus, the exact value of the definite integral is 3 53 ln 3 − 9 26 . The answer is in the form a ln b + c , where a = 3 53 , b = 3 , and c = − 9 26 .
Examples
Imagine you're designing a sound system where the intensity of sound increases logarithmically with distance. Calculating the total sound energy within a specific range involves integrating a function similar to the one in this problem. By finding the definite integral, you can determine the exact sound energy output, which is crucial for optimizing speaker placement and ensuring consistent audio quality throughout the space.
