What is the integral of (5-3x^2)ln(x) ?

The integral of (5-3x^2)ln(x) is 5xln(x)-5x-x^3ln(x)+(x^3)/3+C

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integral of (5-3x^2)ln(x)

\int (5 - 3 x^{2}) ln (x) d x

5 x ln (x) - 5 x - x^{3} ln (x) + \frac{x ^{3}}{3} + C

Solution steps

\int (5 - 3 x^{2}) ln (x) d x

Expand

(5 - 3 x^{2}) ln (x) : 5 ln (x) - 3 x^{2} ln (x)

= \int 5 ln (x) - 3 x^{2} ln (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 5 ln (x) d x - \int 3 x^{2} ln (x) d x

\int 5 ln (x) d x = 5 (x ln (x) - x)

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

= 5 (x ln (x) - x) - (x^{3} ln (x) - \frac{x ^{3}}{3})

Simplify

5 (x ln (x) - x) - (x^{3} ln (x) - \frac{x ^{3}}{3}) : 5 x ln (x) - 5 x - x^{3} ln (x) + \frac{x ^{3}}{3}

= 5 x ln (x) - 5 x - x^{3} ln (x) + \frac{x ^{3}}{3}

Add a constant to the solution

= 5 x ln (x) - 5 x - x^{3} ln (x) + \frac{x ^{3}}{3} + C

f (x) = arctan (x^{2})

\frac{x ^{5}}{5} + \frac{1}{12 x ^{3}}

\int \frac{3 x ^{5}}{1 - 6 x ^{6}} d x

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

f (x) = \frac{3 - x e ^{x}}{x + e ^{x}}

What is the integral of (5-3x^2)ln(x) ?
The integral of (5-3x^2)ln(x) is 5xln(x)-5x-x^3ln(x)+(x^3)/3+C