What is the integral from 1 to e of (4x+2/x)ln(x) ?

The integral from 1 to e of (4x+2/x)ln(x) is e^2+2

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integral from 1 to e of (4x+2/x)ln(x)

\int_{1}^{e} (4 x + \frac{2}{x}) ln (x) d x

e^{2} + 2

Solution steps

\int_{1}^{e} (4 x + \frac{2}{x}) ln (x) d x

Expand

(4 x + \frac{2}{x}) ln (x) : 4 x ln (x) + \frac{2 ln ( x )}{x}

= \int_{1}^{e} 4 x ln (x) + \frac{2 ln ( x )}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{1}^{e} 4 x ln (x) d x + \int_{1}^{e} \frac{2 ln ( x )}{x} d x

\int_{1}^{e} 4 x ln (x) d x = e^{2} + 1

\int_{1}^{e} \frac{2 ln ( x )}{x} d x = 1

= e^{2} + 1 + 1

Simplify

= e^{2} + 2

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What is the integral from 1 to e of (4x+2/x)ln(x) ?
The integral from 1 to e of (4x+2/x)ln(x) is e^2+2