Question:
Integrate the function: \(x log \ 2x\)
Integrate the function: \(x log \ 2x\)
Updated On: Oct 4, 2023
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Solution and Explanation
Let I=∫x log 2x dx
Taking log 2x as first function and x as second function and integrating by parts, we obtain
I = log 2x∫x dx-∫{(\(\frac {d}{dx} 2log \ x\))∫x dx} dx
I = log 2x . \(\frac {x^2}{2}\) - ∫\(\frac {2}{2x}\) . \(\frac {x^2}{2}\)dx
I = \(\frac {x^2log\ 2x}{2}\) - ∫\(\frac {x}{2}\)dx
I = \(\frac {x^2log\ 2x}{2}\) - \(\frac {x^2}{4}\) + C
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The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
