What is the integral of (xln(x))/x ?

The integral of (xln(x))/x is xln(x)-x+C

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integral of (xln(x))/x

\int \frac{x ln ( x )}{x} d x

x ln (x) - x + C

Solution steps

\int \frac{x ln ( x )}{x} d x

Apply u-substitution

: \int e^{u} u d u

= \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= e^{u} u - \int e^{u} d u

\int e^{u} d u = e^{u}

= e^{u} u - e^{u}

Substitute back

u = ln (x)

= e^{l n (x)} ln (x) - e^{l n (x)}

Simplify

e^{l n (x)} ln (x) - e^{l n (x)} : x ln (x) - x

= x ln (x) - x

Add a constant to the solution

= x ln (x) - x + C

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