What is the integral of xlog_{3}(x^2) ?

The integral of xlog_{3}(x^2) is x^2log_{3}(x)-(x^2)/(2ln(3))+C

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integral of xlog_{3}(x^2)

\int x lo g_{3} (x^{2}) d x

x^{2} lo g_{3} (x) - \frac{x ^{2}}{2 ln ( 3 )} + C

Solution steps

\int x lo g_{3} (x^{2}) d x

Simplify

lo g_{3} (x^{2}) : 2 lo g_{3} (x)

= \int 2 x lo g_{3} (x) d x

Take the constant out:

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

= 2 \cdot \int x lo g_{3} (x) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} lo g_{3} (x) - \int \frac{x}{2 ln ( 3 )} d x

= 2 (\frac{1}{2} x^{2} lo g_{3} (x) - \int \frac{x}{2 ln ( 3 )} d x)

\int \frac{x}{2 ln ( 3 )} d x = \frac{x ^{2}}{4 ln ( 3 )}

= 2 (\frac{1}{2} x^{2} lo g_{3} (x) - \frac{x ^{2}}{4 ln ( 3 )})

Simplify

2 (\frac{1}{2} x^{2} lo g_{3} (x) - \frac{x ^{2}}{4 ln ( 3 )}) : x^{2} lo g_{3} (x) - \frac{x ^{2}}{2 ln ( 3 )}

= x^{2} lo g_{3} (x) - \frac{x ^{2}}{2 ln ( 3 )}

Add a constant to the solution

= x^{2} lo g_{3} (x) - \frac{x ^{2}}{2 ln ( 3 )} + C

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What is the integral of xlog_{3}(x^2) ?
The integral of xlog_{3}(x^2) is x^2log_{3}(x)-(x^2)/(2ln(3))+C