Evaluate. (Assume $x>0$.) Check by differentiating.
$\int 5 x \ln \sqrt[4]{x} d x$
$\int 5 x \ln \sqrt[4]{x} d x=$
Answer (1)
Simplify the integrand using the logarithm property: ln 4 x = 4 1 ln x .
Apply integration by parts to evaluate ∫ x ln x d x , choosing u = ln x and d v = x d x .
Substitute the result back into the original expression and simplify.
The final result of the integration is: 8 5 x 2 ln x − 16 5 x 2 + C .
Explanation
Problem Analysis We are asked to evaluate the indefinite integral ∫ 5 x ln 4 x d x , assuming that 0"> x > 0 , and to check the result by differentiating.
Simplifying the Integrand First, we simplify the integrand using the logarithm property ln a b = b ln a . In our case, 4 x = x 4 1 , so ln 4 x = ln x 4 1 = 4 1 ln x . Therefore, the integral becomes ∫ 5 x ln 4 x d x = ∫ 5 x ⋅ 4 1 ln x d x = 4 5 ∫ x ln x d x
Applying Integration by Parts Next, we evaluate the integral ∫ x ln x d x using integration by parts. The formula for integration by parts is ∫ u d v = uv − ∫ v d u . We choose u = ln x and d v = x d x . Then, we find d u = x 1 d x and v = 2 1 x 2 .
Evaluating the Integral Substituting these into the integration by parts formula, we get ∫ x ln x d x = ( ln x ) ( 2 1 x 2 ) − ∫ 2 1 x 2 ⋅ x 1 d x = 2 1 x 2 ln x − 2 1 ∫ x d x = 2 1 x 2 ln x − 2 1 ⋅ 2 1 x 2 + C = 2 1 x 2 ln x − 4 1 x 2 + C
Substituting Back Now, we substitute this result back into the original integral expression: 4 5 ∫ x ln x d x = 4 5 ( 2 1 x 2 ln x − 4 1 x 2 ) + C = 8 5 x 2 ln x − 16 5 x 2 + C
Final Result Thus, the indefinite integral is ∫ 5 x ln 4 x d x = 8 5 x 2 ln x − 16 5 x 2 + C
Checking by Differentiating To check our result, we differentiate it with respect to x :
d x d ( 8 5 x 2 ln x − 16 5 x 2 + C ) = 8 5 ( 2 x ln x + x 2 ⋅ x 1 ) − 16 5 ( 2 x ) = 4 5 x ln x + 8 5 x − 8 5 x = 4 5 x ln x = 5 x ⋅ 4 1 ln x = 5 x ln 4 x
Verification The derivative of our result matches the original integrand, so our integration is correct.
Examples
Imagine you're calculating the total work done by a motor where the force applied changes logarithmically with displacement. This integral helps determine the accumulated work, crucial for designing efficient mechanical systems and predicting energy consumption in engineering applications.
