Final Answer
$\frac{1}{4}$
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Step-by-step Solution
Specify the solving method
1
Simplifying
$\int_{\frac{1}{2}}^{\frac{e}{2}}\frac{\left(1-\ln\left(2x\right)\right)^3}{x}dx$
Learn how to solve definite integrals problems step by step online.
$\int_{\frac{1}{2}}^{\frac{e}{2}}\frac{\left(1-\ln\left(2x\right)\right)^3}{x}dx$
Learn how to solve definite integrals problems step by step online. Integrate the function ((1-ln(2x))^3)/x from 1/2 to e/2. Simplifying. Rewrite the integrand \frac{\left(1-\ln\left(2x\right)\right)^3}{x} in expanded form. Expand the integral \int_{\frac{1}{2}}^{\frac{e}{2}}\left(\frac{1}{x}+\frac{-3\ln\left(2x\right)}{x}+\frac{3\ln\left(2x\right)^2}{x}+\frac{-\ln\left(2x\right)^3}{x}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. Simplify the expression inside the integral.
Final Answer
$\frac{1}{4}$
Exact Numeric Answer
$0.25$