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$\int_{\frac{1}{2}}^{\frac{e}{2}}\frac{\left(1+\ln\left(2x\right)\right)^3}{x}dx$
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. We can solve the integral \int_{\frac{1}{2}}^{\frac{e}{2}}\frac{\ln\left(2x\right)^3}{x}dx by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that 2x it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part.