What is the integral of x*log_{10}(1/x) ?

The integral of x*log_{10}(1/x) is -1/2 x^2log_{10}(x)+(x^2)/(4ln(10))+C

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integral of x*log_{10}(1/x)

integral

x \cdot lo g_{10} (\frac{1}{x})

- \frac{1}{2} x^{2} lo g_{10} (x) + \frac{x ^{2}}{4 ln ( 10 )} + C

Solution steps

\int x lo g_{10} (\frac{1}{x}) d x

Simplify

x lo g_{10} (\frac{1}{x}) : - x lo g_{10} (x)

= \int - x lo g_{10} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int x lo g_{10} (x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} lo g_{10} (x) - \int \frac{x}{2 ln ( 10 )} d x

= - (\frac{1}{2} x^{2} lo g_{10} (x) - \int \frac{x}{2 ln ( 10 )} d x)

\int \frac{x}{2 ln ( 10 )} d x = \frac{x ^{2}}{4 ln ( 10 )}

= - (\frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )})

Simplify

= - \frac{1}{2} x^{2} lo g_{10} (x) + \frac{x ^{2}}{4 ln ( 10 )}

Add a constant to the solution

= - \frac{1}{2} x^{2} lo g_{10} (x) + \frac{x ^{2}}{4 ln ( 10 )} + C

\frac{1}{( x ( ln ( x ) ) )}

u^{- 1}

\frac{( x ) ^{\frac{1}{2}}}{( a - x ) ^{\frac{1}{2}}}

4 y + 1

sec^{2} (y) tan (tan (y))

What is the integral of x*log_{10}(1/x) ?
The integral of x*log_{10}(1/x) is -1/2 x^2log_{10}(x)+(x^2)/(4ln(10))+C