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

The integral of (x^2-1)ln(x) is 1/3 x^3ln(x)-xln(x)-(x^3)/9+x+C

Popular >

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

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

\frac{1}{3} x^{3} ln (x) - x ln (x) - \frac{x ^{3}}{9} + x + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{1}{3} (x^{2} - 3) d x = \frac{x ^{3}}{9} - x

= ln (x) (\frac{x ^{3}}{3} - x) - (\frac{x ^{3}}{9} - x)

Simplify

ln (x) (\frac{x ^{3}}{3} - x) - (\frac{x ^{3}}{9} - x) : \frac{1}{3} x^{3} ln (x) - x ln (x) - \frac{x ^{3}}{9} + x

= \frac{1}{3} x^{3} ln (x) - x ln (x) - \frac{x ^{3}}{9} + x

Add a constant to the solution

= \frac{1}{3} x^{3} ln (x) - x ln (x) - \frac{x ^{3}}{9} + x + C

\int \frac{3 x + 11}{( x ^{2} - x - 6 )} d x

\int x - \frac{1}{3 x} d x

\int sin (8 x) sin (2 x) d x

\int sin (x) \cdot cos (x) \cdot e^{s i n (x)} d x

\int (e^{x} \cdot cos (e^{x})) d x

What is the integral of (x^2-1)ln(x) ?
The integral of (x^2-1)ln(x) is 1/3 x^3ln(x)-xln(x)-(x^3)/9+x+C