Integrate the function: \(x log \ 2x\)
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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Concepts Used:
Integration by Partial Fractions
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,
