What is the integral of (x^2(3ln(x)-1))/((ln(x))^2) ?

The integral of (x^2(3ln(x)-1))/((ln(x))^2) is (x^3)/(ln(x))+C

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integral of (x^2(3ln(x)-1))/((ln(x))^2)

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

\frac{x ^{3}}{ln ( x )} + C

Solution steps

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

Expand

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

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

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

\int \frac{x ^{2}}{ln ^{2} ( x )} d x = - \frac{x ^{3}}{ln ( x )} + 3 Ei (3 ln (x))

= 3 Ei (3 ln (x)) - (- \frac{x ^{3}}{ln ( x )} + 3 Ei (3 ln (x)))

Simplify

= \frac{x ^{3}}{ln ( x )}

Add a constant to the solution

= \frac{x ^{3}}{ln ( x )} + C

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What is the integral of (x^2(3ln(x)-1))/((ln(x))^2) ?
The integral of (x^2(3ln(x)-1))/((ln(x))^2) is (x^3)/(ln(x))+C